Assuming that the denominator is positive, the ratio of a concave and a convex function is typically quasiconcave. That means you can’t use it in CVX without some change, but in certain models, you can make it work. For instance, f(x)/g(x) \geq \alpha becomes f(x) \geq \alpha g(x), which is valid if \alpha>0.
Quasiconvex and quasiconcave objectives can be handled by solving a sequence of convex feasibility problems. Consult Boyd & Vandenberghe for details.