Fun fact, chokidar is a Sanskrit word meaning watchman. It has an interesting root, it comes from the Proto Indo-European word for "four", because he must watch any enclosure of four walls. This ancient word is the source of the Latin "quartus" which comes into English as "quarter" The PIE sense is echoed a little in phrases like "living quarters", "give no quarters" etc
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.