I've never really heard that either, though I've heard "dust-up" before, but that's more of a brawl than a break up. To be honest, I can't really think of much else for a break-up fight, specifically. It may be just because I'm feeling dull-witted right now.
They had big row over it.
They had it out with each other.
They got into an argument.
They had a big episode last week and now they won't speak to each other.
I would have picked the obvious "bust up" in the sense of "break up with," but I had to vote for "bust out" in the sense of "break out (of)." If a prisoner "busts out" of prison he must certainly be severing his relationships with anyone he leaves behind.