PSoTD

I always thought that saying had some sort of hidden meaning, but apparently it's so obvious I really didn't need to look it up:

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, American. 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:

"Unless men can work [the gold] on 'the quiet', they are not likely to make 'piles' so rapidly as Messrs. Hartley and Riley."