I think you are overreaching a bit. The term 'quarter' in 'give no quarter' derives from the French 'quartier', which referred (in military jargon) to a section of a city or military encampment. Prisoners would be kept in 'quartier de sauveté' and 'donner quartier' is short for 'donner quartier de sauveté'. So I disagree that this is an echo of the PIE meaning. The most likely semantic development is that 'quartier', originally a quarter of something (the fourth part), came to denotate any fraction of a whole, and then was specialized in the military lexicon.