The four-node quasi-conforming shell element was extended in the present article to the case of geometrically non-linear behavior of the FGM plates and shells. The high stress occurring in the FGM structures will affect its integrity and the structures is susceptible to failure. Therefore, we focus on the effect of volume fraction of the constituent materials in the mechanical behavior of FGM plates and shells. The material properties are assumed to be varied in the thickness direction according to a sigmoid function in terms of the volume fraction of the constituents. The series solutions of sigmoid FGM (S-FGM) plates, based on the first-order shear deformation theory and Fourier series expansion are provided as the reference solution for the numerical results. In quasi-conforming formulation, the tangent stiffness matrix is explicitly integrated. This makes the element computationally efficient in the non-linear analysis. Several selected examples of non-linear analysis of FGM shells are included in the article for the illustration of possibilities of the presented formulation. It is seen that the present results for the non-linear behavior of FGM plates and shells can provide a useful benchmark to check the accuracy of related numerical solutions.