Give no quarter
Show no mercy or concession; in its original usage, show no mercy for a vanquished opponent.
'Quarter' has long been used to mean 'exemption from being immediately put to death granted to a vanquished opponent by the victor in a battle or fight'. That is exemplified in the definition in Randle Cotgrave's A dictionarie of the French and English tongues, 1611:
Quartier - quarter, or faire war, wherein souldiers are taken prisoners and ransomed at a certaine rate.
To 'give no quarter' derives from this wasn't a phrase welcome to the ears of anyone captured as, 'no quarter given' was effectively a death sentence.
Clearly, the derivation of the phrase depends entirely on the derivation of 'quarter'. Regrettably, this is rather tortuous and, it must be admitted, littered with leaps of the imagination - what the less charitable might call guesses. Nevertheless, let's have a stab. The word quarter could have arrived as part of this idiom by one of two routes. The first route is the longest, so we'll go that way first. The oldest usage of 'quarter' in English is that of our most commonly used present day meaning, that is, a fourth part of something. This usage began in the 13th century as a measure of quantity - being the fourth part of some other measure, for example, eight bushels made one quarter and four quarters made one chaldron (cauldron). By the 14th century, 'quarter' was also beginning to be used as a verb; for example, Robert Manning of Brunne's Langtoft's Chronicle, circa 1338, described some poor victim who was unfortunate enough to be hanged, drawn and quartered:
His hede yei of smyten, to London was it born; ye dede body ye britten [cut into pieces] on four quarters torn.
Approximately, another century on and the sky and space was notionally split into four divisions called quarters. An example of that comes in Geoffrey Chaucer's A treatise on the Astrolabe, circa 1391:
The foure principales plages or quarters of the firmament.
This led on to the four points of the compass being called quarters and, subsequently, districts of a town or city being called quarters. Tyndale's Bible, 1526, records this usage, in Luke XIV 21:
Goo out quickly into the stretes and quarters of the citie.
Of course, we still retain that usage and most large cities have a Latin quarter, a Chinese quarter etc. Along with that gradual modification of meaning there was a parallel development on a domestic scale and a part of a house also became called a quarter. The Paston Letters, 1448 records this:
They han made wyketis [small doors] on euery quarter of ye hwse.
So, we now arrive at a part of a dwelling house being called a quarter, or quarters. That's the first route done with and the notion of captured combatants being granted life and given somewhere to stay, that is, quartered', as opposed to 'given no quarter', that is, killed, is a plausible one.
The second route is straightforward in that 'quarter' was also used to mean 'good relations with or fair treatment of'. This usage dates from around the same time as Cotgrave's 'ransomed from battle' definition and could equally be the source of the meaning of 'give no quarter'; for example, Ralphe Hamor's A True Discourse of the Present Estate of Virginia, 1614:
[The Indians] would without delay dispatch messengers to [their King], to know his purpose and pleasure, desiring faire quarter some 24 howers.
So, after all of the above, we have two quite plausible derivations of the 'quarter' in 'give no quarter'. It may well be that, at this remove, it can't be narrowed down any further than that.
See also: 'at close quarters'.