means A quarter is one of four equal parts of something, or specifically a U.S. coin worth twenty-five cents — a fourth of a dollar.

from From Latin 'quartus,' meaning fourth, by way of Old French 'quartier.' The math is right there in the bones: to quarter is to split into four, whether you're dividing a pie, a city (think the French Quarter, a distinct district of a town), or a dollar. The mercy sense — 'to give quarter' to a defeated enemy — likely comes from the same root, possibly the idea of providing quarters, a place to be housed and spared rather than slain.