This is not much help but I got it from the Askjeeves site.
“Q.T.” certainly looks as though it be the initial letters of two words, every slang authority seems to agree that its origin is simply the first and last letters of the word “quiet.”
The only really surprising thing about “Q.T.” is how long it’s been around. The earliest citation in the Oxford English Dictionary for “Q.T.” is from 1884, in the common form “on the strict q.t.,” which is also found in James Joyce’s “Ulysses” of 1922.
Of course, one would be remiss if one didn’t reference Robin Williams:
“Excuse me, sir, seeing as how the VP is such a VIP, shouldn’t we keep it on the QT, because if it leaks to the VC, he could end up MIA, then we’d all be on KP.”