I found an earlier (1874) appearance of _Q.T._ that shows its origin is most likely from the British side of the pond. This clip appears in an American publication of short plays, but this particular farce, _My Husband's Secret_ by Walter Devereux Whitty, Esq., shows on its title page (scroll up four pages) that it played first in London.
 Here's the [clip][1]: 

![http://books.google.com/books?id=8YtYAAAAMAAJ&pg=PA30-IA27&img=1&zoom=3&hl=en&sig=ACfU3U2SFG975KYWR8jC5_saZwK1lyp8oA&ci=28%2C134%2C444%2C216&edge=0][2]

A note on the date: Google Books lists this publication's date at 1876. The play's title page has its copyright date as 1873. I cross-checked the date of the play at [Open Library][3], where it has 1874.


  [1]: http://books.google.com/books?id=8YtYAAAAMAAJ&printsec=frontcover#v=onepage&q=q.t.&f=false
  [2]: https://i.sstatic.net/Nael6.png
  [3]: http://openlibrary.org/books/OL13830821M/My_husband%27s_secret