For a line bundle L on a smooth projective surface X and nonnegative integers \(k_1, \ldots , k_N\) , Okounkov (Funct Anal Appl 48:138–144, 2014) introduced the reduced generating series \(\big \langle \textrm{ch} _{k_1}^{L} \cdots \textrm{ch} _{k_N}^{L} \big \rangle '\) for the intersection numbers among the Chern characters of the tautological bundles over the Hilbert schemes of points on X and the total Chern classes of the tangent bundles of these Hilbert schemes. Qin (Presentation at the SQuaRE Workshop “Moduli of sheaves on surfaces via Bridgeland stability”, American Institute of Mathematics, San Jose, 2022) conjectured that these reduced generating series are quasi-modular forms if the canonical divisor of X is numerically trivial. In this paper, we verify that Qin’s conjecture holds for \(\langle \textrm{ch} _1^{L_1}\textrm{ch} _1^{L_2} \rangle '\) . The main approaches are to use the methods laid out in Qin and Yu (Intern Math Res Notices 2018:321–361, 2018) and construct various relations regarding multiple q-zeta values and quasi-modular forms.
