An accurate solution of the wave equation at a fluid-solid interface requires a correct implementation of the boundary condition. Boundary conditions at acousto-elastic interface require continuity of the normal component of particle velocity and traction, whereas the tangential components vanish. The main challenge is to model interface waves, namely the Scholte and leaky Rayleigh waves. In this study, we use a nodal discontinuous Galerkin (dG) finite-element method with the medium discretized using an unstructured uniform triangular meshes. The natural boundary conditions in the dG method is implemented by 1) using an explicit upwind numerical flux and 2) by using an implicit penalty flux and setting the modulus of rigidity of the acoustic medium to zero. The accuracy of these methods are evaluated by comparing the numerical solutions with analytical ones, with source and receiver at and away from the interface. The study shows that the solution obtained from the explicit and implicit boundary conditions produces the correct results. This is due to the fact that the stability of the dG scheme is determined the numerical flux, which also implements the boundary conditions by unifying the numerical solution at shared edges of the elements in an energy stable manner.