Oprea and Pandharipande studied the virtual Euler characteristic of Quot schemes on surfaces. Based on the calculations in several cases, they conjectured the rationality of the generating series of the virtual invariants. In this talk, I will explain the virtual $\chi_{-y}$-genera of Quot schemes, thus refining the work of [OP]. The main result expresses the Quot scheme invariants universally in terms of Seiberg-Witten invariants of D"urr, Kabanov, and Okonek. Based on the calculations in several cases and the blow up formula, we resolve the $\chi_{-y}$-genus analogue of the rationality conjecture of [OP], for all surfaces with $p_g>0$. In addition, the reduced Quot scheme invariants for $K3$ surfaces will be discussed with connections to the Kawai-Yoshioka formula.