To "show no quarter" or announce "no quarter given" is a figure of speech
meaning "show no mercy
." It usually is used when referring to battle, where showing no quarter
meant killing the enemy instead of capturing them.
Quarter means "place or area," and not necessarily one fourth of anything. It was used in names of places like the "Latin Quarter," and "quarters" were barracks for soldiers, as in the phrase quartering of troops. To "grant quarter" meant "to provide a prisoner with shelter," which of course meant not to kill him. Denying "quarter" originally literally meant denying shelter, then changed over time into denying the right of survival.
Ancient history contains some examples of generals ordering troops to spare soldiers defeated in battle, and others like the Roman legion massacring all males in a city. However, there were also attempts by early lawgivers to punish those judged too brutal with the enemy. Muhammad (pbuh), for example, forbade the killing of prisoners of war, explicitly instructing the fighters to treat them as if they were one of their family. Modern efforts to ensure humane treatment of enemy fighters date back to the American Civil War, when President Abraham Lincoln put into law in 1863 what became known as the Lieber Code, a codification of the laws of armed conflict, which expressly forbade Union troops to give no quarter. This was a precursor to the Geneva Convention. A conference in Brussels in 1874, attended by European powers in the aftermath of the Franco-Prussian war, eliminated what had been a major loophole in the Lieber Code, the so-called "great straits" provision, which allowed a soldier or a fighting unit to eliminate survivors of battle when their “own salvation” made taking prisoners impossible.
In non-battle terms, the term No Quarter could mean to not co-operate or concede, drive a hard bargain.
If your enemy announces that he will grant no quarter, then to you it's a fight to the death. Before the Battle of the Alamo, Santa Anna ordered that a red flag be raised indicating to the defenders that no quarter would be given. All the men who weren't killed in the battle were subsequently captured and executed. So it goes.
The phrase is not all that common, but it still gets circulated; "No quarter for jerks," "No quarter given to terrorists," "No quarter will be given to Canada in hockey," etc.