division vs quartering

division and quartering both are nouns.

Word	Noun	Adjective	Verb	Adverb
division	Yes	No	No	No
quartering	Yes	No	No	No

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

division (noun)	quartering (noun)
the act or process of dividing	dividing into four equal parts
an arithmetic operation that is the inverse of multiplication; the quotient of two numbers is computed	living accommodations (especially those assigned to military personnel)
the act of dividing or partitioning; separation by the creation of a boundary that divides or keeps apart	a coat of arms that occupies one quarter of an escutcheon; combining four coats of arms on one shield usually represented intermarriages
one of the portions into which something is regarded as divided and which together constitute a whole
discord that splits a group
an army unit large enough to sustain combat
a group of ships of similar type
a unit of the United States Air Force usually comprising two or more wings
an administrative unit in government or business
(botany) taxonomic unit of plants corresponding to a phylum
(biology) a group of organisms forming a subdivision of a larger category
a league ranked by quality