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:
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, where it has 1874.