Xenon tetrafluoride (XeF₄), a square-planar molecule with the D_4h point group, exhibits complex vibrational properties owing to anharmonicity, vibrational coupling, and symmetry-induced degeneracies. In the current study, its vibrational spectra were viewed with a symmetry-adapted U(2) Lie algebraic perspective. The vibrational frequencies of molecular vibrations are defined by the coupled local anharmonic oscillators related to the Xe–F stretching and the F–Xe–F bending modes. A symmetry-specific vibrational Hamiltonian is derived using Casimir and Majorana operators to account for anharmonic effects, bond interactions, and vibron-exchange processes that preserve the molecular symmetry D_4h. The Hamiltonian matrix is defined in a local basis and diagonalized to yield vibrational eigenvalues for fundamental, first, and second overtone vibrations. The predicted vibrational frequencies fit the observed values rather well, thereby validating the proposed model. The XeF₄ study findings detail vibrational delocalization, anharmonic corrections, and symmetry-induced splitting. The results indicate that the symmetry-adapted U(2) Lie algebraic framework serves as an effective tool for providing a reasonable description of the vibrational spectra of highly symmetric polyatomic molecules, while preserving the essential symmetry properties and anharmonic characteristics of the system.