2 companions that dislike each other np if both have one companion they like.
but if you have one companion that dislikes 2 other companions and likes one, he will leave.
2 lords that dislike eachother? Talk to your wife "resolve dispute" quest
Then the "ex" companion (now lord) is no longer in your party and they will no longer have any fights going on. I do remember something from when I didn't know which companions where noble that their disputes and friendschips continued as vasals, example: alayen supported ymira, and deshavi had a fight with borcha. This doesn't seem to be a coincidence.
In any case, for 2 people to have a fight, they have to be or in your party (both) or both your vassals.