In the scalar case, you’re all set, using CVX’s rel_entr
rel_entr(rho,zrho)
In the matrix case, I think https://en.wikipedia.org/wiki/Trace_inequalities#Joint_convexity_of_relative_entropy shows the convexity of your expression, presuming that I am correctly interpreting the article as applying to the matrix log, as opposed to element level log, and presuming that rho and zrho are both semidefinite. I think what you need in this case is not support for logm, but for quantum relative entropy, which perhaps might also be known as matrix relative entropy.
mcg can provide the definitive prognosis when he comes around.