quarter (n.) Look up quarter at Dictionary.com
c.1300, "one-fourth of anything," from Old French quartier (12c.), from Latin quartarius "fourth part," from quartus "fourth" (see quart). Earliest sense is "parts of the body as dismembered during execution" (c.1300). Used of the moon from c.1400 and the hour from 1590s. The coin is peculiar to U.S., first recorded 1783.

Meaning "region, locality" is from c.1300. Meaning "portion of a town" (identified by the class or race of people who live there) is first attested 1520s. Quarter days (late 15c.), when rents were paid and contracts and leases began or expired, were, in England, Lady day (March 25), Midsummer day (June 24), Michaelmas day (Sept. 29), and Christmas day (Dec. 25); in Scotland, keeping closer to the pagan Celtic calendar, they were Candlemas (Feb. 2), Whitsunday (May 15), Lammas (Aug. 1), and Martinmas (Nov. 11). Quarter horse, bred strong for racing on quarter-mile tracks, first recorded 1834.
quarter (v.) Look up quarter at Dictionary.com
"to cut in quarters," early 15c., from quarter (n.). Related: Quartered; quartering. The verb meaning "to put up soldiers" is recorded from 1590s (see quarters).