A novel implicit cell-vertex finite volume method is presented for the solution of both the Navier-Stokes equations and the governing equations for certain viscoelastic fluids. The key idea is the elimination of the pressure term from the momentum equation by multiplying the momentum equation with the unit normal vector to a control volume boundary and integrating thereafter around this boundary. The resulting equations are expressed solely in terms of the velocity components and, where appropriate, the components of elastic stress. Thus any difficulties with pressure or vorticity boundary conditions are circumvented and the number of primary variables that need to be determined equals the number of space dimension. As test cases, lid-driven cavity flow in a square enclosure and flow around a confined circular cylinder in a channel are solved in order to verify the accuracy of the present method and extensive comparisons are made with the results available in the literature. In addition, we investigate some interesting flow instabilities such as asymmetric flow around a confined circular cylinder at high blockage ratios and the stabilization effect of viscoelasticity for this flow. The numerical results are computed on meshes having up to 1.8 million degrees of freedom.