  We define and study the space of $q$-opers associated with Bethe equations
for integrable models of XXZ type with quantum toroidal algebra symmetry. Our
construction is suggested by the study of the enumerative geometry of cyclic
quiver varieties, in particular, the ADHM moduli spaces. We define
$(\overline{GL}(\infty),q)$-opers with regular singularities and then, by
imposing various analytic conditions on singularities, arrive at the desired
Bethe equations for toroidal $q$-opers.
