  We extend equivariant dimensional reduction techniques to the case of quantum
spaces which are the product of a Kaehler manifold M with the quantum
two-sphere. We work out the reduction of bundles which are equivariant under
the natural action of the quantum group SU_q(2), and also of invariant gauge
connections on these bundles. The reduction of Yang-Mills gauge theory on the
product space leads to a q-deformation of the usual quiver gauge theories on M.
We formulate generalized instanton equations on the quantum space and show that
they correspond to q-deformations of the usual holomorphic quiver chain vortex
equations on M. We study some topological stability conditions for the
existence of solutions to these equations, and demonstrate that the
corresponding vacuum moduli spaces are generally better behaved than their
undeformed counterparts, but much more constrained by the q-deformation. We
work out several explicit examples, including new examples of non-abelian
vortices on Riemann surfaces, and q-deformations of instantons whose moduli
spaces admit the standard hyper-Kaehler quotient construction.
