[laurent_du]

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.

[jbreckmckye]

Well, it's uncertain that "quarter" in this sense came from the Middle French. Early Modern English also uses "quarter" to mean "the relations between people", Shakespeare uses it for instance

The actual link is obscure, but I don't think it matters too much - I'm observing a similarity, not attesting an origin.