A second-order accurate cell-vertex finite volume/finite element hybrid scheme is proposed. A finite volume method is used for the hyperbolic stress equations and a finite element method for the balance equations. The finite volume implementation incorporates the recent advancement on fluctuation distribution schemes for advection equations. Accuracy results are presented for a pure convection problem, for which fluctuation distribution has been developed, and an Oldroyd-B benchmark problem. When source terms are included consistently, second-order accuracy can be achieved. However, a loss of accuracy is observed for both benchmark problems, when the flow near a boundary is (almost) parallel to it. Accuracy can be recovered in an elegant manner by taking advantage of the quadratic representations on the parent finite element mesh. Compared to the finite element method, the second-order accurate finite volume implementation is 10 times as efficient.