"open place where rocks are excavated," c. 1400 (mid-13c. as a place name), from Medieval Latin quareia, a dissimilation of quarreria (mid-13c.), literally "place where stones are squared," from Latin quadrare "to make square," related to quadrus "a square," quattuor "four" (from PIE root *kwetwer- "four").

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 "the fourth, fourth part," which is related to quattuor "four," from PIE root *kwetwer- "four." 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.

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 horse, bred strong for racing on quarter-mile tracks, first recorded 1834. 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.

also quarter-staff, 1540s (quarter-stroke "stroke with a quarterstaff" is attested from early 15c.), stout pole, six to eight feet long (six-and-a-half sometimes is given as the standard length), tipped with iron, formerly a weapon used by the English peasantry. From staff (n.); the quarter is of uncertain signification. According to one theory, favored by fencing manuals, etc., it likely is in reference to operation of the weapon:

It was grasped by one hand in the middle, and by the other between the middle and the end. In the attack the latter hand shifted from one quarter of the staff to the other, giving the weapon a rapid circular motion, which brought the ends on the adversary at unexpected points. [Century Dictionary]

Linguists tend to prefer an explanation from woodcutting, perhaps a reference to a cut of lumber known as a quarter, but contemporary evidence is wanting for either conjecture.