The crossing numbers of a graph is a vital parameter and a hard problem in the forefront of topological graph theory.  Determining the crossing number of an arbitrary graphs is NP-complete problem. Because of its difficultly, the classes of graphs whose crossing number have been determined are very scarce. In this paper, for the special graph Q on six vertices, we through the disk drawing method to prove that the crossing numbers of its join with n isolated vertices as well as with the path P_{n} and with the cycle C_{n} are cr(Q+nK_{1})=Z(6,n)+2\lfloor\frac{n}{2}\rfloor, cr(Q+P_{n})=Z(6,n)+2\lfloor\frac{n}{2}\rfloor+1 and cr(Q+C_{n})=Z(6,n)+2\lfloor\frac{n}{2}\rfloor+3, respectively.