Residence or place of abode; as, winter quarters, the place where an army lodges during the winter months. We say “this quarter of the town,” meaning this district or part; the French speak of the Latin Quartier- i.e. the district or part of Paris where the medical schools, etc., are located; the Belgians speak of quartiers a $$$ lodgings to let; and bachelors in England often say, “Come to my quarters”- i.e. apartments. All these are from the French verb écarter (to set apart).
“There shall no leavened bread be seen with thee, neither shall there be leaven seen ... in all thy quarters [any of thy houses].” - Exodus xiii. 7.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894