An adaptive moving mesh finite difference scheme is developed for tokamak magneto-hydrodynamic (MHD) simulations, based on the CLT code (S. Wang and Z.W. Ma, Phys. Plasmas, 2015). Our numerical scheme is built on the MHD equations in curvilinear coordinates, based on a coordinate transformation from the physical domain to a computational domain. The scheme is constructed on a uniform Cartesian computational mesh that is obtained from a non-uniform adaptive moving mesh in the physical domain through the coordinate trans-formation. Mesh points in the physical domain in general move and concentrate in the vicinity of solutions with rapid variations by solving an adaptive mesh equation, whilst total number of mesh points remains unchanged. The local resolution can be significantly increased and computational resource is largely reduced. Comparison between results obtained with the original uniform mesh and the new adaptive moving mesh is carried out by simulation of the linear and nonlinear 2/1 tearing mode, linear and nonlinear 1/1 resistive internal kink mode. It is found that the adaptive moving mesh scheme possesses better numerical stability and convergence.