I know this question, how to take all the shots set in the bracketing with one shutter press, has been asked several times but the answer has been CL or CH or Self Timer. But while I was playing around, I've set it to Q and all three shots fires with single shutter press, however, it is not consistent. When I try again at another instance, this might not work. Is there some other settings involved to make this happen again or is it a flaw in the software?
I have tested this with my own D7200 and had the same result you described.
After talking to a Nikon tech, he explained me that the camera is doing this in the quiet mode by only taking up the mirror once and doing the rest with the curtain. (The clack you hear when pressing the shutter is the mirror flipping up. The curtain is very quiet.)
It works normally as you expect it when you switch it to S mode. It is not a flaw in the software. It is what Nikon wanted so it is more quiet.