There are two parts to your idiom, it seems to me.
1) That even the closest people eventually fight and
2) That because they are close, they eventually work things out
While there are idiomatic expressions in English that might mean one or the other, I can't think of any that simultaneously mean both.
For (2), as has already been suggested, "blood is thicker than water", meaning family ties will overcome any extra-familial disturbances or loyalties. Another one would be "family is forever" which has different shades of meaning but one being that eventually you have to work out problems with these people because you are stuck with them. A similar one would be "you can't pick your family" or "you can pick your friends, but you can't pick your family," meaning you must find some way to get along with these people, because you have no other choice.
For (1), there is a subtle difference between EVEN people who are very close eventually fight (let's call this 1a) and BECAUSE people are very close they eventually fight (1b). I can't think of a good expression for (1a), but there are plenty for (1b): "familiarity breeds contempt", "no one is a prophet in their own land", and "too much kin and less than kind".