A couple of places to start for further looking:
The paper Further summation formulae related to generalized harmonic numbers by Zheng mentions the $p = q = 2$ case in example 2.3, and has several related results, but not explicitly of the form you're looking for.
Appendix B of the paper On some log-cosine integrals related to $\zeta(3), \zeta(4)$ and $\zeta(6)$ by Mark Coffey has some similar things, and maybe looking through its references and papers that cite it would yield more.
