"indulgence or mercy shown to a vanquished foe; exemption from being immediately killed upon being defeated in battle or armed contest," 1610s, presumably from quarter (n.1) or its French equivalent quartier in this meaning, but OED writes that "the precise origin of this sense is obscure ...." It suggests quarter in a now-obsolete sense of "relations with, or conduct towards, another" (attested from 1640s), or possibly quarter in the sense of "place of stay or residence" (compare quarters), on the notion of sending the vanquished to an assigned place until his fate is decided. German quartier, Swedish quarter, Danish kvarteer, etc. probably are from French.
mid-15c., "day that begins a quarter of the year," designated as days when rents were paid and contracts and leases began or expired, from quarter (n.1). They 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 pre-Christian 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. Related: Quarter days.
As a verb, "to play quarterback," by 1945. The figurative sense (the quarterback when on the field typically directed the team's plays) is from 1952. Monday morning quarterback is by 1932 as a noun, "second-guesser of other people's decisions, one who criticizes or passes judgment afterward on how something was done;" by 1972 as a verb; originally pro football player slang for sportswriters (U.S. professional football games typically were played on Sundays).
1620s, "the part of the spar-deck of a man-of-war between the poop and the main-mast," originally "a smaller deck above the half-deck," covering about a quarter of the vessel [OED], from quarter (n.1).
"It is used as a promenade by the officers only" [Century Dictionary], hence the colloquial nautical noun quarter-decker (by 1867) "an officer who is more looked upon as a stickler for small points of etiquette than as a thorough seaman."
early 15c., quarterli, "four times a year, once a quarter," from quarter (n.1) + -ly (2). As an adjective from mid-15c., "occupying alternate quarters" (of a coat of arms), with -ly (1). As a noun, "a quarterly publication," from 1830, from the adjective. Earlier the adverb was used in a now-obsolete sense of "into quarters" (c. 1400).
also quarter-master, early 15c., "subordinate officer of a ship," from French quartier-maître or directly from Dutch kwartier-meester; originally a ship's officer whose duties included stowing of the hold; later (c. 1600) an officer in charge of quarters and rations for troops. See quarters.
"military dwelling place," 1590s, from quarter (n.1) in sense of "portion of a town." As "part of an American plantation where the slaves live," from 1724. The military sense seems to be also the source of quartermaster and it might be behind the phrase give quarter "spare from immediate death" (1610s, often in the negative), on the notion of "provide a prisoner with shelter;" see quarter (n.2).
also quarter-staff, 1540s (quarter-stroke "stroke with a quarterstaff" is attested from early 15c.), an old weapon formed from a stout pole, six to eight feet long (six-and-a-half sometimes is given as the standard length), tipped with iron, formerly a weapon characteristic of the English peasantry. From staff (n.); the quarter in it 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.
also quartette, 1773, "musical composition for four solo instruments or voices," from French quartette, from Italian quartetto, diminutive of quarto "fourth," from Latin quartus "the fourth, fourth part" (related to quattuor "four," from PIE root *kwetwer- "four"). Meaning "set of four singers or musical players who perform quartets" is from 1814.
mid-15c., originally in astrology and astronomy in the phrase quartile aspect in reference to celestial bodies when 90 degrees apart in longitude; see quartile (adj.). In statistics, from 1879.