quarter (n.) Look up quarter at Dictionary.com
c.1300, "one-fourth of anything; one of four parts or divisions of a thing;" often in reference to the four parts into which a slaughtered animal is cut, from Old French quartier, cartier (12c.), from Latin quartarius "fourth part," from quartus "fourth" (see quart). One of the earliest dated references in English is to "parts of the body as dismembered during execution" (c.1300).

Used of the phases of the moon from early 15c. The use of quarter of an hour is attested from mid-15c. In Middle English quarter also meant "one of the four divisions of a 12-hour night" (late 14c.), and the quarter of the night meant "nine o'clock p.m." (early 14c.).

From late 14c. as "one of the four quadrants of the heavens;" hence, from the notion of the winds, "a side, a direction" (c.1400). In heraldry from mid-14c. as "one of the four divisions of a shield or coat of arms." The word's connection with "four" loosened in Middle English and by 15c. expressions such as six-quartered for "six-sided" are found. Meaning "region, locality, area, place" is from c.1400. Meaning "portion of a town" (identified by the class or race of people who live there) is first attested 1520s. For military sense, see quarters. As a period of time in a football game, from 1911. Quarter horse, bred strong for racing on quarter-mile tracks, first recorded 1834.

The coin (one fourth of a dollar) is peculiar to U.S., first recorded 1783. But quarter could mean "a farthing" in Middle English (late 14c.), and compare British quadrant "a farthing" (c.1600), and classical Latin quadrans, the name of a coin worth a quarter of an as (the basic unit of Roman currency).

Quarter days (mid-15c.), designated as days 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 in the sense "period of three months; one of the four divisions of a year" is recorded from late 14c.
quarter (v.) Look up quarter at Dictionary.com
"to cut in quarters, divide into four parts," mid-14c., from quarter (n.). Specifically as the word for a form of criminal punishment from late 14c. (Old English had slitcwealm "death by rending"). Related: Quartered; quartering. The meaning "to put up soldiers" is recorded from 1590s (see quarters).