A few weeks ago I was chatting with some friends after karate class and the usage of the phrase "on the qt" came up. Surprisingly to me, a couple of my friends hadn't even heard of it and thought it was actually supposed to be "on the dl". Now that was a phrase I had never heard. Apparently, it means "on the down low" - i.e. hush hush. And this basically is the same meaning as "on the qt."
But, we wondered, what does "qt" stand for? Well, I finally just remembered the conversation and used that helpful intarweb thang and came across this link. So, it appears "qt" is an abbreviation for "quiet". The phrase has been around since the late 1800's, too! I guess that's why only my older friends seemed to know what it meant. :-0