Let H be a fixed k-vertex graph with m edges and minimum degree d > 0. We use the learning graph framework of Belovs to show that the bounded-error quantum query complexity of determining if an n-vertex graph contains H as a subgraph is O(n^[2 - 2/k -t] ), where t = max {A, B} > 0 with A = [ k^2 -2(m+1)] / [k(k+1)(m+1)] and B = [2k -d -3]/[k(d+1)(m-d+2)]