This paper is focused on a ratio-dependent predator-prey model with cross-diffusion of quasilinear fractional type. By applying the theory of local bifurcation, it can be proved that there exists a positive non-constant steady state emanating from its semi-trivial solution of this problem. Further based on the spectral analysis, such bifurcating steady state is shown to be asymptotically stable when the cross diffusion rate is near some critical value. Finally, numerical simulations and ecological interpretations of our results are presented in the discussion section.