How to get a 1 clock period pulse from a constant input?
If the input is asynchronous with respect to the clock, it can't be done with a single flip-flop.

It's easy to show that an abstract state machine that implements the specification needs to have at least three states:


Before the pulse
During the pulse
After the pulse


You can't implement a 3-state machine with only a single flip-flop.

If the input is synchronous with the clock, then that needs to be stated as part of the problem. Then you can use the input signal directly to generate one of the output pulse edges, as in your Figure 1.