The “patch test” verifies whether a linear solution is reproduced exactly in an elasticity problem. This approach to test the numerical formulation and the code itself is standard in the finite element method. The MLS shape functions do not have a polynomial form. Therefore, the integration is not well performed by the classical Gauss–Legendre scheme and the patch test is only satisfied asymptotically at convergence. In this paper, we propose a custom quadrature scheme for MLS shape functions in order to ensure the properties needed for an exact verification of the patch test.