  We determine the quantum multiplication with divisor classes on the Hilbert
scheme of points on an elliptic surface $S \to \Sigma$ for all curve classes
which are contracted by the induced fibration $S^{[n]} \to \Sigma^{[n]}$. The
formula is expressed in terms of explicit operators on Fock space. The
structure constants are meromorphic quasi-Jacobi forms of index $0$. Combining
with work of Hu-Li-Qin, this determines the quantum multiplication with
divisors on the Hilbert scheme of elliptic surfaces with $p_g(S)>0$. We also
determine the equivariant quantum multiplication with divisor classes for the
Hilbert scheme of points on the product $E \times \mathbb{C}$.
  The proof of our formula is based on Nesterov's Hilb/PT wall-crossing, a
newly established GW/PT correspondence for the product of an elliptic surface
times a curve, and new computations in the Gromov-Witten theory of an elliptic
curve.
