  We compute the Small Quantum Cohomology of Hilb^2(P^1xP^1) and determine
recursively most of the Big Quantum Cohomology. We prove a relationship between
the invariants so obtained and the enumerative geometry of hyperelliptic curves
in P^1xP^1. This extends the results obtained by Graber for Hilb^2(P^2) and
hyperelliptic curves in P^2.
