The transition-metal pentatelluride $\mathrm{HfT}{\mathrm{e}}_{5}$ exhibits unique topological properties and anomalous transport behaviors at low temperatures, with the underlying physical mechanism hypothesized to be linked to the formation of charge density waves (CDWs). However, current experimental studies lack direct evidence to confirm the presence of CDWs in $\mathrm{HfT}{\mathrm{e}}_{5}$. Here, using ultrafast coherent phonon spectroscopy, we investigate the bosonic quasiparticle dynamics of $\mathrm{HfT}{\mathrm{e}}_{5}$ and identify two low-energy coherent oscillations in the sub-THz range (\ensuremath{\sim}0.55 THz and \ensuremath{\sim}0.72 THz), which show significant frequency softening as the critical temperature (${T}^{*}\ensuremath{\sim}75\phantom{\rule{0.16em}{0ex}}\mathrm{K}$) is approached. Excluding the possibility of acoustic phonons through pump wavelength- and fluence-dependent time-domain oscillation analysis, we propose that these modes are likely collective excitations of amplitude modes of CDWs, driven by a minimally nested wave vector in $\mathrm{HfT}{\mathrm{e}}_{5}$. Our findings provide experimental evidence for the potential existence of CDW order in $\mathrm{HfT}{\mathrm{e}}_{5}$ from a nonequilibrium perspective, which may offer insight into the microscopic origins of its exotic physical properties.