How do I implement the delay of the Q-Phase Input in GMSK?
A system that shifts the input signal by 90 degrees is a Hilbert transformer. For a single sine wave, a 90 degree phase shift is simple (e.g., sine becomes cosine), but for a general signal you need a Hilbert transformer. Check the basics here.
It can be implemented using an FIR (finite impulse response) digital filter. Such a filter can be designed in Matlab/Octave.

EDIT: The above is an answer to the question "how do I perform a $90^{\circ}$ phase shift (for non-sinusoidal signals)?". However, what I didn't notice - and what was luckily pointed out by Jim Clay in a comment - is the fact that in the given diagram there is no need for a phase shifter. The first $90^{\circ}$-block simply symbolizes the splitting of the signal in an I-component and a Q-component. It is not a phase splitter and the $90^{\circ}$-symbol is actually quite misleading.