The flow and diffusion of miscible fluid in a porous medium with a high Plcet number (Pe) and large viscosity ratio widely exist in industrial processes, such as oil recovery, geological sequestration of carbon dioxide, and chemical engineering process. When these problems are studied by numerical methods, the key point is to accurately describe the flow dynamics and diffusion process in a porous medium at the same time. As an alternative to conventional numerical methods, the lattice Boltzmann method based on kinetic theory is well suited to pore-scale simulations of miscible fluid flows and molecular diffusion. However, most of the existing lattice Boltzmann models have many difficulties (e.g. robustness and numerical stability) in simulating such systems at high Pe and large viscosity ratio. In this paper, in order to overcome the above difficulties, we propose a coupled lattice Boltzmann model based on the multiple-relaxation-time model and the lattice kinetic scheme for the fluid flow and diffusion, respectively. It can be shown that the incompressible Navier-Stokes equations and the convection-diffusion equation can be derived from the presented coupled model through the Chapman-Enskog procedure. The proposed model is validated by simulating a concentration gradient driven flow in a porous channel. Numerical results demonstrate that the model is of second-order accuracy in space. We further simulate a flow through two types of artificial porous media. The robustness of the presented model is investigated by measuring the permeability and diffusivity under different relaxation times. It is found that the model is insensitive to relaxation parameters. In addition, the miscible viscous displacement in two parallel plates is simulated to test the numerical stability of the model. It is observed that the results accord well with those reported in previous work, and the model is very stable at high Pe and large viscosity ratio in comparison with the standard lattice Bhatnagar-Gross-Krook model. Overall, the coupled lattice Boltzmann model can serve as an effective tool for directly simulating the fluid flow and diffusion at high Pe and large viscosity ratio in the pores of a porous medium.