The nonconforming triangular piecewise quadratic finite element space by Fortin and Soulie is used to approximate the elliptic obstacle problem in 2D. A priori error estimate to this solution is derived to show that the convergence of the error behaves optimal.   The flux of the problem is reconstructed in the H(div,\Omega)-conforming finite element spaces, namely Brezzi-Douglas-Marini (BDM_1) and Raviart-Thomas (RT_1). A  reliable a-posteriori error estimates based on those reconstructed fluxes is derived. This can be tracked to a result  by Prager and Synge. Moreover, various numerical examples have been discussed to show  the optimal convergence of  the adaptive finite element method (AFEM). We demonstrate   the nonconforming quadratic FEM is an efficient method to  deal with  the obstacle problem for  the elastic membrane.