  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.
