A theoretical study of the electronic and nuclear flux densities of a vibrating H2+ molecular ion is presented. The time-dependent wave function is represented in the basis of vibronic eigenstates which are numerically obtained from the complete nonrelativistic Hamiltonian without the clamped-nuclei approximation. A one-center expansion in terms of B-splines and Legendre polynomials is employed to solve the corresponding eigenvalue equation. The electronic and nuclear flux densities are then calculated from the total wave function through their quantum-mechanical definition. Analysis of the flux densities close to the turning points shows that the nuclear wave packet takes longer time (1.4 fs) to change its direction compared to the electronic one (1 fs).