That's a very difficult and situational question to answer. In some cases, yes, this done happen. In others, no. It all depends on how the bird bonds and with whom. If a bird decides to see someone or some bird as a mate, they become very, very protective and territorial over them. If they see them as a flock mate, then it's more like they're buddies.
There are many cases when the opposite of what you suggest happens, where the birds hate each other because one of them has chosen you as their mate. I don't believe this behaviour is always preventable, but you can at least work towards trying to prevent it through proper socialization and not allowing one bird too much time with one person or bird. Though, I do imagine it is much more difficult with birds that are naturally monogamous. Birds that are promiscuous (it sounds funny, but it's the proper term) will not choose one mate. Instead, they love everyone and do not discriminate, though they may occasionally choose a favourite for a short period of time.
My GCC, Avery, will randomly show protective "mate" behaviour over me. When she's with me, she may lunge at my boyfriend or friends when they get too close and get full of attitude with them (it's really quite funny!). But I don't encourage this behaviour with her, and I also don't let her on my shoulder when others are around as a general rule with her, as she shows it more the "closer" she is to me. When she does do this, I actually put her down and have that person use some target training or bribe her with a banana or sunflower seed to step up on them and get close to them. It doesn't always work, but I am at least showing her that being aggressive to people around me doesn't give her power over whom I choose to be with.