Consider the following idioms.
To push one's buttons. Meaning: to do things that create a very strong emotional reaction in one, especially anger, irritation, or exasperation:
I hate Mary's new boyfriend, he's always trying to push my buttons, and he's doing a good job of it!
To hit/strike/touch a nerve. Meaning: to make someone feel angry, upset, embarrassed, etc.:
Something she said to him must have hit/struck/touched a nerve. I've never seen him so angry.
So, the idioms above can all be used to describe hurting someone's feelings. However, they don't mean a fight per se. For example, if you say "they are pushing each other's buttons" or "They are touching a nerve," it doesn't necessarily meaning having a fight. But it means hurting, possibly during an argument:
I hate to fight with her! She knows how to push my buttons/hit a nerve because her words hurt like a knife!