The slang term 'qt' is a shortened form of 'quiet'. There's no definitive source for the phrase "on the qt", although it appears to be of 19th century British origin. The longer phrase 'on the quiet' is also not especially old, but is first recorded somewhat before 'on the qt', in Otago: Goldfields & Resources, (1862) which reads, "Unless men can work [the gold] on 'the quiet', they are not likely to make 'piles' so rapidly as Messrs. Hartley and Riley."

That first record is from new Zealand, but is soon followed by citations from the United Kingdom and the USA.

As to on the q.t., in The Facts on File Encyclopedia of Word and Phrase Origins, Robert Hendrickson states, "A British broadside ballad (1870) contained the line 'Whatever I tell you is on the Q.T.'"

It would be good to know the name of the ballad in order to follow up this assertion. Unfortunately, the author doesn't give it, from which we can only suppose he didn't know it himself. Without some supporting evidence that claim has to be in doubt.