Coming from a nautical background, I believe this term to be derived in part from the days of the 'men of war', or sailing vessels. I currently don't recall the name of the relatively infamous skipper who used the phrase, but what it refers to is the quarter of a ship -- the port and starboard sides aft, just forward of the transom.
The cannons on these fighting ships were usually placed along the sides, the stern section housing officer quarters. So, to say you will give or show no quarter, is to basically say you will not allow the ass end of your vessel to line up with the guns of your foe. Such condition would leave your vessel most vulnerable.
It is interesting how I used the word quarters to describe the officer berths above. This made me think, as I'm sure the meaning of this term also implies "take no prisoners". This same meaning could easily have come from the days of sail as well.
The Captain exulted, "I WILL SHOW YEE NO QUARTER!", as the salvo of cannon shot bore through his enemy's waterline.