In order to solve the plane unsteady and quasi-static coupled thermoelasticity problems with concave convex boundary shape, the element free Galerkin method (EFG) is used. The subsequent results are adaptively optimized by using the moving least square method (MLS) to construct the shape function and the Lagrange multiplier method to deal with the essential boundary conditions (the first kind of boundary conditions), as well as introducing the Voronoi adjacency criterion and the posteriori error formula. Then a new EFG adaptive model for unsteady quasi-static and coupled thermoelasticity problems is constructed. To verify the model’s feasibility, the temperature field and displacement field distribution in the planes with smooth and concave convex boundary shape are calculated under two-dimensional mixed boundary conditions. The results are compared with those of finite element method. The difference between the results of finite element method and meshless method is characterized, and the effectiveness and accuracy of EFG for unsteady quasi-static thermoelasticity coupled problems are verified.