The Hilbert-Chow map from the Hilbert scheme of points on a surface to the symmetric product of the surface is a contraction with the extremal ray generated by a rational curve. We will discuss the computation of the Gromov-Witten invariants of the Hilbert scheme with respect to this curve class. The main techniques are the cosection method of Kiem and J. Li and the computation of Gromov-Witten invariants for the Hilbert scheme of the complex plane by Okounkov and Pandharipande.