housing vs quarters

housing and quarters both are nouns.

Word	Noun	Adjective	Verb	Adverb
housing	Yes	No	No	No
quarters	Yes	No	No	No

As nouns, quarters is a hyponym of housing; that is, quarters is a word with a more specific, narrower meaning than housing:

housing (noun)	quarters (noun)
structures collectively in which people are housed	housing available for people to live in
a protective cover designed to contain or support a mechanical component
stable gear consisting of a decorated covering for a horse, especially (formerly) for a warhorse