Toward Qin's Conjecture on Hilbert schemes of points and quasi-modular forms

For a line bundle $L$ on a smooth projective surface $X$ and nonnegative integers $k_1, \ldots, k_N$, Okounkov \cite{Oko} introduced the reduced generating series $\big \langle {\rm ch}_{k_1}^{L} \cdots {\rm ch}_{k_N}^{L} \big \rangle'$ 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. In \cite{Qin2}, Qin 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 $\langle {\rm ch}_1^{L_1}{\rm ch}_1^{L_2} \rangle'$. The main approaches are to use the methods laid out in \cite{QY} and construct various relations regarding multiple $q$-zeta values and quasi-modular forms.