We investigate the dynamics of a quantized vortex in a trapped superfluid Fermi gas near the unitarity limit. By taking a trial wave function for the order-parameter of a condensate in a rotating axisymmetric trap confinement and using a time-dependent variational analysis we obtain the equations of motion and their solutions for anomalous mode. The results show that the critical rotating frequency of the trap increases when the system ranges from the left to the right side of the unitarity limit, while the period of the vortex decreases in this regime.