quarter

Quarter comes from Middle English quarter, borrowed via Anglo-Norman quarter from Latin quartarius, itself from quartus, meaning 'fourth.' Spanish cuarto, meaning 'room' or 'quarters,' comes from the same root.

The facts here don't detail a specific shift in meaning over time, so the honest story is simply that the word has carried its 'fourth part' sense from Latin through Anglo-Norman into English.

Quarter is a doublet of quartier — the two words descend from the same source but arrived in English by different paths.