My dictionary (OED) says "quarrel" is commonly constructed with "over", "about", and "for". Actually, however, "quarrel for" seems to be of rare use. As for the choice between "over" and "about", the dictionary suggests nothing, though I personally feel "over" is more common at least in formal writing.
(EX) The two countries have long quarreled over the ownership of tiny islands.
(EX) Oh, please don't quarrel about me. Let me go home, and then all will be OK.
(EX) They quarreled for the bread, because each wished to have the largest loaf.
paco