quarrel vs fall out

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

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

As verbs, fall out is a hyponym of quarrel; that is, fall out is a word with a more specific, narrower meaning than quarrel:

Other hyponyms of quarrel include brawl, wrangle, spat, polemicise, polemicize, polemise, polemize.

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

quarrel (verb)	fall out (verb)
have a disagreement over something	come to pass
	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