fall out vs quarrel

fall out is not a noun while quarrel is a noun.

Word	Noun	Adjective	Verb	Adverb
fall out	No	No	Yes	No
quarrel	Yes	No	Yes	No

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

Other hypernyms of fall out include altercate, argufy, dispute, scrap.

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

fall out (verb)	quarrel (verb)
come to pass	have a disagreement over something
have a breach in relations
drop out
leave (a barracks) in order to take a place in a military formation, or leave a military formation
come as a logical consequence; follow logically