We develop and implement a finite volume method based on a conservative approach for the hyperbolic non-conservative system of the shear shallow water model. This model is a generalization of the shallow water model to include the effects of shear, which are ignored in the standard shallow water model. The existing path conservative scheme is extended to unstructured grids composed of triangles or quadrilaterals with the solution stored at the cell centers. The high-resolution scheme is based on a local solution reconstruction idea for which a new set of variables based on a Cholesky decomposition of velocity fluctuation tensor is developed, which is found to increase the robustness of the schemes by maintaining positive definiteness property. The scheme is applied to some test problems like 1D dam break, square dam break, roll waves, and radial hydraulic jump.