debate vs quarrel

debate and quarrel both are nouns.

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

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

debate (noun)	quarrel (noun)
the formal presentation of a stated proposition and the opposition to it (usually followed by a vote)	an arrow that is shot from a crossbow; has a head with four edges
a discussion in which reasons are advanced for and against some proposition or proposal	an angry dispute

debate (verb)	quarrel (verb)
argue with one another	have a disagreement over something
discuss the pros and cons of an issue
have an argument about something
think about carefully; weigh