We introduce a new second-order central-upwind scheme for shallow water equations on the unstructured quadrilateral grids. We propose a new technique for bottom topography approximation over quadrilateral cells as well as an efficient water surface correction procedure which guarantee the positivity of the computed fluid depth. We also design a new quadrature for the discretization of the source term, using which the new scheme exactly preserves “lake at rest” steady states. We demonstrate these features of the new scheme as well as its high resolution and robustness and its potential advantages over the triangular central-upwind scheme in a number of numerical examples.