In this paper, we investigate the shallow water flows over erodible beds by using a fully coupled mathematical model in two-dimensional space. This model combines the nonlinear shallow water equations, the sediment transport equation and the bed evolution equation. The purpose of this paper is to design a well-balanced and positivity-preserving scheme for this model. In order to achieve the well-balanced property, the coupled system is first reformulated as a new form by introducing an auxiliary variable. The central discontinuous Galerkin method is applied to discretize the model. By choosing the value of the auxiliary variable suitably, the scheme can exactly balance the flux gradients and source terms in the "still-water" case, and thus the well-balanced property of the proposed scheme can be proved. Moreover, the non-negativity of the volumetric sediment concentration in the sediment transport equation is maintained by choosing a suitable time step and using a positivity-preserving limiter. Numerical tests are presented to illustrate the validity of the proposed scheme.