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

and the first term produces simply by the definition of expectation value. The problem is to get rid of the in the second expectation value.

To do so, use chapter 12.8, 2. That shows that the second term is essentially modified by factors of the form

where the integration is over the unit sphere. If you use the representation of the spherical harmonics as given in {D.65}, you can relate these inner products to the unit inner products

Have fun.

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.