For a connected graph G, the Schultz and modified Schultz polynomials are defined as, respectively, where the summations are taken over all unordered pairs of distinct vertices in V(G), is the degree of vertex u, is the distance between u and v and V(G) is the vertex set of G. In this paper, we find Schultz and modified Schultz polynomials of the Cog-special graphs such as a complete graph, a star graph, a wheel graph, a path graph and a cycle graph. The Schultz index, modified Schultz index and average distance of Schultz and modified Schultz of each such Cog-special graphs are also obtained in this paper.