We explore changes in the adiabatic low-order g-mode pulsation periods of 0.526, 0.560, and 0.729 M$_odot$ carbon-oxygen white dwarf models with helium-dominated envelopes due to the presence, absence, and enhancement of $^{22}$Ne in the interior. The observed g-mode pulsation periods of such white dwarfs are typically given to 6$-$7 significant figures of precision. Usually white dwarf models without $^{22}$Ne are fit to the observed periods and other properties. The root-mean-square residuals to the $simeq$ 150$-$400 s low-order g-mode periods are typically in the range of $sigma_{rms}$ $lesssim$ 0.3 s, for a fit precision of $sigma_{rms}/ P$ $lesssim$ 0.3 %. We find average relative period shifts of $Delta P/P$ $simeq$ $pm$ 0.5 % for the low-order dipole and quadrupole g-mode pulsations within the observed effective temperature window, with the range of $Delta P/P$ depending on the specific g-mode, abundance of $^{22}$Ne, effective temperature, and mass of the white dwarf model. This finding suggests a systematic offset may be present in the fitting process of specific white dwarfs when $^{22}$Ne is absent. As part of the fitting processes involves adjusting the composition profiles of a white dwarf model, our study on the impact of $^{22}$Ne can provide new inferences on the derived interior mass fraction profiles. We encourage routinely including $^{22}$Ne mass fraction profiles, informed by stellar evolution models, to future generations of white dwarf model fitting processes.
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