  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.
