We present a numerical scheme for modeling unresolved turbulence in cosmological adaptive mesh refinement codes. As a first application, we study the evolution of turbulence in the intracluster medium (ICM) and in the core of a galaxy cluster. Simulations with and without subgrid scale (SGS) model are compared in detail. Since the flow in the ICM is subsonic, the global turbulent energy contribution at the unresolved length scales is smaller than 1% of the internal energy. We find that the production of turbulence is closely correlated with merger events occurring in the cluster environment, and its dissipation locally affects the cluster energy budget. Because of this additional source of dissipation, the core temperature is larger and the density is smaller in the presence of SGS turbulence than in the standard adiabatic run, resulting in a higher entropy core value.