This means "I have a dispute or unpleasant matter to settle with you".
A similar Irish phrase is "I have a crow to pluck with you".
It seems to be a reference to picking the last bits of meat off a bone. This is a difficult business; the implication is that settling the dispute will mean a lot of difficult and detailed discussion.
Hi, there's also a simliar idiom in my mother language, "pick a bone from an egg", it means someone's picky. but "have a bone to pick with sb" is obviously different in the meaning.Just for your information.