What does Boxing Day have to do with boxing?
c.1200, "free, clear" (of debt, etc.), from Old French quite, quitte "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" (see quiet (adj.)).
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 (see quit (adj.)).
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.