To find how many triangles a graph G has in general you can write down the adjacency matrix A, where A(i,j) = 1 if there is an edge between vertex i and vertex j, and 0 otherwise. Taking the square of the matrix A^2(i,j) counts the number of k such that there is an edge (i,k) and an edge (k,j). So A(i,j).A^2(i,j) counts the number of triangles containing (i,j) as an edge. Sum over all i,j and divide by 3.
However in this case I think it's easier. K(1,n) is the complete bipartite graph on 1+n vertices, say 0,1,...,n with 0 connected to each of 1,...n. So its complement is simply the complete graph on 1,...n. It's easy to see that this has n(n-1)(n-2)/6 triangles.