Quit
Middle English (in the sense ‘set free’): from Old French quiter (verb), quite (adjective), from Latin quietus, past participle of quiescere ‘be still’, from quies ‘quiet’.
wiktionary
From Middle English quiten, quyten, from Anglo-Norman quitter, Old French quitter, from quitte(“acquited, quit”), ultimately from Latin quietus.
Compare Dutch kwijten(“to quit”), German Low German quitten(“to quit”), German quitten, quittieren, Danish kvitte, Swedish qvitta, kvitta(“to quit, leave, set off”), Icelandic kvitta.
Probably of imitative origin.
etymonline
quit (adj.)
c. 1200, "excused, exempt, free, clear" (of debt, obligation, penalty, etc.), from Old French quite, quitte "free, clear, entire, at liberty; discharged; unmarried," and directly 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").
From mid-13c. as "deprived of." From c. 1300 of real property, "exempt from taxes or other dues or claims."
quit (v.)
c. 1200, quiten, "to repay, discharge" (a debt, claim, etc.), from Old French quiter "to clear, establish one's innocence;" also transitive, "release, let go; absolve, relinquish, abandon" (12c., Modern French quitter), 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, repay" is from 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 "to leave, depart from, go away from" is attested by late 14c.; that of "stop, cease" (doing something) is from 1640s. Meaning "to give up, relinquish" is from mid-15c. Related: Quitted; quitting. Quitting time "time at which work ends for the day" is from 1835.
