quarrel vs argue

quarrel is a noun but argue is not a noun.

Word	Noun	Adjective	Verb	Adverb
quarrel	Yes	No	Yes	No
argue	No	No	Yes	No

As verbs, argue is a hypernym of quarrel; that is, argue is a word with a broader meaning than quarrel:

Other hypernyms of quarrel include contend, debate, fence.

quarrel (noun)	argue (noun)
an arrow that is shot from a crossbow; has a head with four edges
an angry dispute

quarrel (verb)	argue (verb)
have a disagreement over something	have an argument about something
	give evidence of
	present reasons and arguments