Numerical simulation of morphodynamic problems is considered. The physical model is based on the shallow-water equations coupled with the Exner equation closed by the Grass model to describe the time evolution of the bed profile. The SRNH predictor-corrector scheme and a modified Roe scheme for non-conservative systems of equations are considered for space discretization. Second-order accuracy in space is achieved through variable reconstruction. These schemes were previously used in the simulation of the considered problems together with explicit time advancing. Linearized implicit time-advancing versions are generated here, in which the flux Jacobians are computed through automatic differentiation. Second-order accuracy in time is obtained through a backward differentiation formula associated with a defect-correction approach. For both the considered numerical methods, the explicit and implicit versions are compared in terms of accuracy and efficiency for one-dimensional and two-dimensional morphodynamic problems characterized by different time scales for the evolution of the bed and of the water flow.