We establish in this paper the supercloseness of the quadratic finite element solution of a two dimensional elliptic problem to the piecewise quadratic interpolation of its exact solution. The assumption is that the partition of the solution domain is quasi-uniform under a Riemannian metric and that each pair of the adjacent elements in the partition forms an approximate parallelogram. This result extends our previous one in [7] for the linear finite element approximations based on adaptively refined anisotropic meshes. It also generalizes the results by Huang and Xu in [13] for the supercloseness of the quadratic elements based on the mildly structured quasi-uniform meshes. A distinct feature of our analysis is that we transform the error estimates on each physical element to that on an equilateral standard element, and then focus on the algebraic properties of the Jacobians of the affine mappings from the standard element to the physical elements. We believe this idea is also useful for the superconvergence study of other types of elements on unstructured meshes.