The collective excitations of a one-dimensional superfluid Fermi gas in an anharmonic trap are investigated. By using the variational approach, the frequency shifts about the dipole mode of the center-of-mass variations and the breathing mode of width variations are derived. It is found that the frequency shift in a unitary region is more significant. Under the excitations of different driving amplitudes, the two low-energy modes are coupled due to the contribution of the quartic item, and the quantum beating phenomenon comes into being. The frequency of beating increases with the driving amplitude. The dynamics of the width exhibits complex characteristics, especially, in the unitary region.