Short summary yet very long document. Sorry if I forgot anything. This is for those who want to know stuff about the lore but won't go looking for it.

The order of the agents is in the same order they joined the valorant protocol. I've done everything, but I'm posting up to Cypher because of how long it got. If you guys want, I'll post the rest.

Brimstone:

• Viper: he definitely knew her before Valorant protocol. He respects her search for revenge. Upon killing her, there’s a line where he says he hopes she’ll find peace. He also has a line to an enemy Viper in which he says they didn’t need to be enemies. He also calls her Sabine in a more recent voiceline. Viper calls him old friend as well and upon killing him, tells him to rest. Their relationship is pretty solid and good.
• Killjoy: definitely knew her as well and likes her very much. It’s confirmed that he’s her guardian. When killing her, he says “sorry kid” or “you left me no choice, kid”. Killjoy returns the feelings. She says it feels bad to think of killing the enemy Brimstone and asks the team Brimstone to watch over them. They have a really good relationship.
• Omen: Generic voice lines. It’s not possible to say whether Brimstone knew Omen or if he knows who Omen was before becoming a phantom. Omen also doesn’t have any voice lines that implicate he knows Brimstone very well.
• Cypher: Only one generic voiceline. Cypher also doesn’t have a very specific voice line. He only suggests hacking into his bracelet.
• Sova: Brimstone says Sova lost an eye for them and that’s enough. He also calls him “soldier” which means a deep respect coming from a veteram like him. Sova returns the feeling. He likes Brimstone and seems to like to hear his stories.
• Sage: Brimstone considers her to be second in command of the Valorant protocol. He has a line where he says that if he dies, she takes the lead, and she has the same line, saying she’ll take the lead if he dies.
• Phoenix: Brimstone thinks Phoenix brags too much. Phoenix makes fun of his age. Overall, they don’t sound like enemies, nor good friends either.
• Jett: Only mentions she should stretch before running. Jett mentions his age and acknowledges his leadership.
• Reyna: Fears reyna, basically. Says he wants to keep his soul. Reyna criticizes his morality and is reluctant to accept his leadership.
• Raze: Generic voice lines about Raze destroying things. Raze complains that Brimstone won’t let her listen to music. There’s also a line where she asks Killjoy to go “tinkering” with her and Killjoy says that if something blows up, she’ll be the one telling Brimstone. So it’s safe to assume they are good allies.
• Breach: doesn’t seem to like Breach at all. Even when Breach is in his team, he says “You did well, Breach, don’t expect a hug”. Breach is very, very annoyed by Brimstone. But he has a line saying that it could be worse than working together: facing each other. So they do have differences, being forced to tolerate the other. (Edit) Breach accuses Brimstone of protecting Kingdom in a new voiceline. Brimstone says he has no idea what he's talking about. Perhaps their beef comes from Brim working for Kingdom in the past while Breach has always been against them.
• Skye: I don’t recall a single voice line directed at her. She basically makes fun of his age.
• Yoru: I don’t recall a single voice line directed at him either. He makes fun of his age.
• Astra: Doesn’t have voice lines directed at her yet. She calls him boomer.

Viper:

• Omen: She seems to be the only person who knows who Omen was before becoming what he is. She knows his real name, calls him old friend, but apparently withholds information from him. When she lets his name slip, she quickly stops herself. Apparently, Omen can’t know who he was. Omen also knows Viper real name, and never uses the codename Viper. He seems nostalgic of the past with her. When he kills her, he says “experiment complete” for some reason. He mentions that he and Viper became monsters, but she still has flesh to hide her horrors. They seem to have the most complex relationship between two agents.
• Killjoy: Surprisingly, Viper seems to respect Killjoy a lot. She says the enemy Killjoy is too smart for her own good so she has to reprimand her. Killjoy also seems to respect her very much. Upon seeing an enemy Viper, Killjoy says it’s a shame to rid the world of her mind, but she needs to rest. When Viper is in her team, she says she’ll “do the tests today” because Viper is biased, no offence (because of her hatred). Viper also calls Killjoy “little mouse”. In german, using animal names for people you care about is very common.
• Cypher: She doesn’t like that Cypher knows her real name and demands to be called “Doctor Sabine” if he insists on it. She never demanded such a thing from Omen or Brimstone. Cypher teases her about knowing someone’s birthday on the 20th of March. They have an overall neutral-ally relationship.
• Sova: Generic voice lines. It’s hard to make up what they think of each other. Sova comments on how violent she is and wonders if there’s another way to find peace.
• Sage: that’s a complex one. Viper is pissed at her. She says Sage failed her. Sage has a recent voice line where she asks Viper if she thought about her offer, and Viper cuts her mid sentence to claim she shouldn’t even dream of helping her, because she can’t help her, and she can’t help “them”. Upon killing her, Viper says “that’s payback, Sage”.
• Phoenix: Generic lines about burning and stealing oxygen.
• Jett: She seems to like Jett but their age difference is very discrepant. Viper comments on Jett’s youth a lot. Jett doesn’t say much about Viper.
• Reyna: Viper comments that Reyna is just as vengeful as herself. Reyna loves that Viper is just as hateful as she is. Despite that, they don’t seem very close.
• Raze: Admits that Raze’s tinkering is good, but not openly. Raze calls her on a “girls only” night, which is pretty funny. Seems like they are good allies, but Raze doesn’t know the limits of their relationship.
• Breach: Viper has a line where he says they hired Breach. She also tells him to help them breach the enemy open. Despite Brimstone’s annoyance, Viper could be responsible for him being in the Valorant protocol. When Breach kills Viper, he says it’s not personal, just business, further confirming this.
• Skye: I don’t recall voice lines towards her, but Skye doesn’t like the poison in the air, alluding to pollution.
• Yoru: I don’t recall any voice lines towards him. He says she needs to die.
• Astra: A recent tweet in the german Valorant page says she absolutely hates Viper. There are no lines directed at Astra yet.

Omen:

• Killjoy: Generic. Says he’ll break her mind. She’s also very generic towards him.
• Cypher: Asks if he has seen under his hood. Weird. Cypher comments that he has the information Omen asked him, but it will come at a price.
• Sova: Interestingly, Sova says Omen will find the answer he’s looking for, and thus he won’t be this way forever, which means Omen might go back at being Fred. Omen is generic towards him.
• Sage: Interesting voice lines. He asks if Sage knows where the life she gives comes from. When Sage is in the opposite team, he says there’s a radiant healer tipping the balance. Sage comments on the pain he feels.
• Phoenix: Funniest voice line in the game. He says Omen knits. Omen says it helps him focus. That’s pretty much it.
• Jett: Very generic. Omen praises her like “well done”.
• Reyna: Omen claims he can hear the souls Reyna steals. Reyna praises Omen. But Reyna likes everyone who’s a radiant, so that’s not a big deal.
• Raze: Omen comments on Raze being loud and misses laughing like she does. Raze is scared by him and is thankful that he’s on their side (when he is).
• Breach: Calls Omen a freak. Rude. Omen just says he’ll silence him.
• Skye: Omen calls her a force of nature. There’s a line in which he wonders “what she would’ve become” but says it doesn’t matter now. Not sure what that means. But he admires her powers. Skye is confused by his teleport in at least two voice lines.
• Yoru: Omen really likes Yoru, probably because their powers are very similar. He asks if Yoru can feel parts of his soul being left behind as he passes through rifts. Yoru likes Omen for the same reasons.
• Astra: no lines towards her. I don’t remember a line towards Omen.

Killjoy:

• Cypher: complains about him stealing stuff from her lab and says his creations are very simple, like the things she used to do in her childhood. But calls him a genius in a voice line. Cypher admits to stealing her stuff and asks for materials. They seem good “spying and hacking” buddies. (Edit thanks to u/ISuckAtGamesHehe for reminding me) In a more recent line, Cypher seems very nice to her when she asks him help with the deep web and he is shocked and worried about "protecting her".
• Sova: She wants his eye to see what it has recorded. Sova is generic towards her.
• Sage: Calls Sage to hear her. I don’t remember Sage mentioning her.
• Phoenix: I don’t remember any voice lines towards him. He calls her tech support.
• Jett: Jett does admire Killjoy and is reluctant to face an enemy Killjoy if she’s as smart as theirs. She says Killjoy’s bots are good and so is she. Killjoy doesn’t mention her, as far as I remember.
• Raze: Killjoy and Raze seem very close to each other. It’s probably the second most context-full relationship in the game, after Viper and Omen. Killjoy calls her “girlfriend” in german (which, just like English, could both mean as in girlfriend or girl friend). She also calls Raze little mouse, which is an affectionate way to talk to your loved ones. Killjoy also says Raze drives her mad and talks a lot about Raze tinkering with her bots, but at the same time says she did a good job on them. Raze calls her to go “tinkering” with her, and Killjoy accepts. I found a trivia in the valorant wiki page saying the bird in Killjoy’s turret might be painted by Raze to look like the Brazilian soccer team mascot, known as “canarinho pistola”. Additionally, Raze says “Killjoy, you and me till the end”. Sweet.
• Breach: he calls her nerd. She tried to improve his arms.
• Reyna: Probably the most obvious antagonist relationships in the game. Reyna absolutely hates Killjoy. Killjoy doesn’t seem to hate her back but fears her. Reyna says she saw what her creations did to someone, then cut herself and won’t say who.
• Skye: they tease each other a lot because one uses nature and the other uses hi tech.
• Yoru: Killjoy thinks he’s a cool kid.
• Astra: no voice lines yet.

Cypher:

• Sova: Cypher has both lines in which he thinks very dearly of Sova and as if they are big enemies. Cypher is probably the only agent Sova doesn’t love.
• Sage: Cypher seems to be interested in Sage’s ability to revive people, probably because she could revive his family. Sage is very generic towards him.
• Phoenix: Generic. The most expressive thing Cypher says to Phoenix is about being scared of how impressive he is. Phoenix is thankful for Cypher’s stuff.
• Jett: He’s annoyed that she breaks stuff with her dash. Jett simply asks where they are so she can kill them.
• Raze: Again, he’s pretty annoyed that she breaks his equipment. Raze is sorry for that.
• Breach: Says Cypher is worst than Interpol. Which means Breach was a hunted criminal before being hired by Viper. He’s annoyed by Cypher’s spying. Cypher has stolen stuff from his arm for his equipment.
• Skye: calls people for a tea at Cypher’s place, but doesn’t know where it is. Praises Cypher. Cypher doesn’t mention her.
• Yoru: Doesn’t like Cypher. Says he’s invisible so he can’t be seen by Cypher. Cypher thinks he’s cool.
• Astra: no voice lines.