Two incompressible SPH numerical solvers, including a modified explicit method and a new implicit method have been developed to simulate the sediment-laden free surface flow problems. Using, consistent discretization schemes, the proposed explicit method improves somewhat the accuracy of the usual explicit ISPH methods. The implicit method additionally guarantees the incompressibility condition completely. In the presented methods, the liquid phase is modeled using Navier-Stokes equations and to predict the non-Newtonian behavior of the sediment phase, the Bingham plastic rheological model is used. The accuracy and capabilities of the developed incompressible SPH methods is first validated in comparison with available experimental and numerical results of a single-phase water-sediment mixture flow generated by unsteady dam break problem. Then, they are applied to simulate an eroding dam break problem with a two-phase flow sediment transport. Comparing the obtained results with the available results in the literature shows that the developed methods particularly the implicit one, are very powerful tools for simulation of the problems including sediment transport induced by violent free surface flow, with interactions between flow and sediment and morphological changes in bed.