The slang term 'qt' is a shortened form of 'quiet'. There's no definitive source for the phrase 'on the q.t.', although it appears to be of 19th century British origin - not, as is often supposed,...
On the q.t. is slang for quiet. q.t.. is an abbreviation for quiet. The phrase dates to at least 1884 when it appears in George Moore's A Mummer's Wife (Source: Oxford English Dictionary, 2nd Edition)