Hilbert schemes of points on surfaces and multiple q-zeta values

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, and conjectured that they are multiple $q$-zeta values of weight at most $\sum_{i=1}^N (k_i + 2)$. The second-named author further conjectured in \cite{Qin2} that these reduced generating series are quasi-modular forms if the canonical divisor of $X$ is numerically trivial. In this paper, we verify these two conjectures for $\big \langle {\rm ch}_2^{L} \big \rangle'$. The main approaches are to apply the procedure laid out in \cite{QY} and to establish various identities for multiple $q$-zeta values and quasi-modular forms.