The aim of this article is to show the global existence of both martingale and pathwise solutions of stochastic equations with a monotone operator, of the Ladyzenskaya-Smagorinsky type, driven by a general Levy noise. The classical approach based on using directly the Galerkin approximation is not valid. Instead, our approach is based on using appropriate approximations for the monotone operator, Galerkin approximations and on the theory of martingale solutions.