bickering vs quarrel

bickering and quarrel both are nouns.

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

As nouns, quarrel is a hypernym of bickering; that is, quarrel is a word with a broader meaning than bickering:

Other hypernyms of bickering include dustup, row, run-in, words, wrangle.

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

bickering (verb)	quarrel (verb)
	have a disagreement over something