Can I output one flip-flop into another on the same clock pulse?
Well, no. You can't clock the output from one DFF into a second DFF on the same rising-edge.

What you can do, though, is clock the 2nd DFF on the falling edge (i.e. invert the clock signal).

Something like this:




This way both flip-flops will hold the same data at the next rising edge.
Just make sure that your clock pulse width is larger than the gate delays + trace delay from Q1 to D2.