This expression means “in confidence” or “just between us” and is originally British. To expand on that a bit, the first citation in the OED is from the Irish writer George Moore’s 1885 novel A Mummer’s Wife: “It will be possible to have one spree on the strict q.t.” And the first citation in Green’s Dictionary of Slang, also from 1885, is some lines of verse from a Sydney, Australia, newspaper: “Oh, my! what a pious world it is, / And how very good they all seem to be – / But what a ’duffing’ lot you’d find / If you would only raise the blind, / And see ’em on the strict Q.T.” (Green’s also has a citation supposedly from an 1870 song, but that is pretty clearly inaccurate.)
In both the OED and Green’s, all the citations through 1904 except one use the formulation “on the strict q.t.” and are from British, Irish, or Commonwealth sources. The one exception (in Green’s) is from a Provo, Utah, newspaper in 1894: “We got this on the dead Q-T —and will ask you readers, please don’t give it away.” And incidentally, the expression gets some literary pedigree via James Joyce’s Ulysses (1922): “Sailing under false colours after having boxed the compass on the strict q.t. somewhere.”
All sources that I’ve found say that “q.t.” is short for “quiet,” and that’s convincing, especially since the phrase “on the quiet,” meaning the same thing, can be found pre-1874. However, I was intrigued by an alternate theory offered by a contributor to the Stack Exchange English Language and Usage bulletin board:
“Q.T.” is an odd abbreviation for “quiet.” Since it is of British origin, I would think it would derive from schoolboys’ abbreviations, often derived from Latin. The Latin taceo means “not to speak” and has solemn meaning sometimes, referring to “passing over in silence.” Thus quae tacenda, or q.t., would refer to “things about which one should not speak.” Cf. Horace, Epodes, 5.49, where Horace speaks of Canidia and quid dixit et quid tacuit, what she said and what she left unsaid.
Another Stack Exchange contributor, who goes by “Callithumpian,” antedated the OED and Green’s initial citations by more than ten years, unearthing a passage from a British play called My Husband’s Secret that debuted no later than 1874:
It quickly became a catchphrase: a Google Books search for “on the strict q.t.” yields about a dozen examples from 1877 to 1880, including these:
The phrase had fully arrived in the U.S. in the early 1900s. Green’s has fourteen citations between 1906 and 1999, and nine of them are American, including this lyric from a song by Merle Travis: “You thought your little romance was on the strict Q.T. / So if you want your freedom P.D.Q., / Divorce me C.O.D.”
The song came out in 1946, and indeed the phrase has a mid-century feel. The 1997 film L.A. Confidential was set in the early ’50s and the gossip monger played by Danny DeVito’s trademark line is “Off the record, on the QT, and very hush-hush.”
Nowadays, “on the q.t.” has been replaced by “on the downlow” or “on the d.l.”–though that phrase also has a very particular meaning of its own.
notoneoffbritishisms.com 
I’ve only read it in British novels, and that only rarely; mostly wartime stuff, said between men.
I have never heard or read it anywhere in Australia, and the vast majority of Australians would have no idea what it meant.
I have only ever heard it used in a jocular fashion, suggesting that it has been considered old-fashioned for many decades. Incidentally, “had fully arrived” sounds quite bizarre to my English ears!
How so?
It’s just a phrase I have never heard anyone say. I suppose that something has either arrived or it hasn’t.
Ah. Now I completely understand.
yeah, definitely a dated feel to that one. likewise “mum’s the word” is old-fashioned as well
I would have assumed this one was British just because it seems older than American. But I guess not. I’ve always felt the expression had the connotation of being somewhat secretive in a gossipy sense, rather than just something regarded as simply confidential, but maybe that’s just me.
The schoolboy Latin explanation seems like one of those clever but undoubtedly false etymological theories (like the many explanations of ps and qs). I draw the opposite conclusion from the mention of Oxford in the play excerpt. As far as I can tell from the one page given, it seems to be a working-class cockney character who knows the phrase, and he is mocking an educated character for not knowing the slang of the common people. He is not suggesting it actually was a phrase from Oxford.