The writ of discharge formerly granted to those barons and knights who personally attended the king on a foreign expedition. At their discharge they were exempt from the claim of scutage or knight's fee. Subsequently the term was applied to the acquittance which a sheriff receives on settling his account at the Exchequer; and, later still, to any discharge of an account: thus Webster says-
“You had the trick in audit-time to be sick till I had signed your quietus.” —Duchess of Malfy (1623).
Who would fardels bear ... When he himself might his quietus make With a bare bodkin?
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894
More on Quietus from Fact Monster: