The heterogeneous elasticity problems existed many material interfaces. While the classical discontinuous Galerkin finite element method faced the unstable numerical problem resulting from the inappropriate stability parameter for elasticity problem with interfaces. This problem could be released by the weighted Nitsche discontinuous Galerkin finite element method for constant elements. The weights and the stabilization parameters of the weighted Nitsche discontinuity Galerkin finite element method were derived with four-node quadrilateral elements discretization, and a qualitative dependence between the weights and the stabilization parameters was established. This made the use of high-order elements possible. The weights and the stabilization parameters were evaluated numerically by setting up and solving generalized eigenvalue problems. The convergence and stability of the proposed method were verified through numerical examples. The results show that the weighted Nitsche discontinuous Galerkin finite element method has good stability and high accuracy for both homogeneous and heterogeneous material problems. It provides a good foundation for further use in discontinuities such as cracking problems.