(noun)
a disagreement, conflict, or quarrel, usually heated and engaged in verbally, and often involving some trivial issue;
e.g. She was engaged in a quarrel with him about whose turn it was to use the room.
(noun)
a reason to complain, a basis for having a disagreement or dispute with someone or something;
e.g. I have no quarrel with his technique, as long as it yields good results.
(verb)
to actively engage in a dispute or disagreement with someone, to heatedly argue with or about something;
e.g. She didn't want to quarrel with them, so she allowed them to continue with what they were doing.
(verb)
to criticize or disagree with something, to find fault in it (usually used as "quarrel with");
e.g. He couldn't quarrel with any part of their statement, so he agreed to grant their request.