c. 1200, "to repay, discharge" (a debt, etc.), from Old French quiter "clear, establish one's innocence;" also transitive, "release, let go, relinquish, abandon" (12c.), from quite "free, clear, entire, at liberty; discharged; unmarried," from Medieval Latin quitus, quittus, from Latin quietus "free" (in Medieval Latin "free from war, debts, etc."), also "calm, resting" (from PIE root *kweie- "to rest, be quiet").
Meaning "to reward, give reward" is mid-13c., that of "take revenge; to answer, retort" and "to acquit oneself" are late 14c. From c. 1300 as "to acquit (of a charge), declare not guilty." Sense of "leave, depart" is attested from c. 1400; that of "stop" (doing something) is from 1640s. Meaning "to give up, relinquish" is from mid-15c. Related: Quitted; quitting. Quitting time is from 1835.