The result for one proton is readily available in literature and messy
to derive yourself. If you want to give it a try anyway, one way is
the following. Note that in spherical coordinates
To do so, use chapter 12.8, 2. That shows
that the second term is essentially
modified by factors of the form
The expression for the quadrupole moment if there are an odd number 3 of protons in the shell would seem to be a very messy exercise. Some text books suggest that the odd-particle shell model implies that the one-proton value applies for any odd number of protons in the shell. However, it is clear from the state with a single hole that this is untrue. The cited result that the quadrupole moment varies linearly with the odd number of protons in the shell comes directly from Krane, [30, p. 129]. No derivation or reference is given. In fact, the restriction to an odd number of protons is not even stated. If you have a reference or a simple derivation, let me know and I will add it here.