It can still be used. But it's much rarer than 'argued' (same meaning) or 'fought' (broader meaning).

Quarrel is less common nowadays, but it is still used.

According to the British National Corpus, the word "quarrel" is the 10,248th most common word where as "argue" is 2363rd. In short, it is much more common. Besides that, "argue" is more rational, reasoned (in theory) while "quarrel" is more emotionally charged.

"Quarreled" is not used in conversational English. You better use argue. However, you might use it in written, like in a narrative or a descriptive essay. It will sound kind of odd if you used it in conversation.