"Give no quarter!" involves combat or struggle, and urges one's own forces to keep on keeping on, without ceding any ground to the enemy's. It has little or nothing to do with human kindness in the execution of an enemy.
After defeating the enemy, the cry "Give no mercy!" has to do with treating an offender or an enemy very harshly, even up to execution. The execution need not be pain-free (cruel and unusual punishment). A more extreme view of "Give no mercy!" would be "Take no enemies!" and indicates one must kill the fallen enemy, for what good would it do to let them escape?