The numerical solutions of time $\alpha$-order $(\alpha \in (0,1))$ Caputo fractional Fokker-Planck equations is considered. The constructed method is consist of the transformed $L1$ ( $TL1$) scheme in the temporal direction and the Legendre-Galerkin spectral method in the spatial direction. It has been shown that the $TL1$ Legendre-Galerkin spectral method in $L^2$-norm is exponential order convergent in space and ( $2-\alpha$)-th order convergent in time. Several numerical examples are given to verify the obtained theoretical results.