quarters

[Tacitus]

We are all familiar with the sense of 'quarters' being fourth parts of something. But how do we relate this to its use as a place in which something happens. As in 'sleeping quarters', 'battle quarters', 'headquarters', 'close quarters', and the verbal use to quarter meaning to put people in to living or sleeping quarters ?

[Perry]

quarters
"military dwelling place," 1590s, from quarter (n.) in sense of "portion of a town." The military sense is in quartermaster (mid-15c.) and might be behind the phrase give (no) quarter (1610s), on the notion of "to provide a prisoner with shelter." The verb quarter "to put up soldiers" is recorded from 1590s.

Online Etymology Dictionary