In the absence of a third dredge-up episode during the asymptotic giant branch phase, white dwarf models evolved from low-metallicity progenitors have a thick hydrogen envelope, which makes hydrogen shell burning be the most important energy source. We investigate the pulsational stability of white dwarf models with thick envelopes to see whether nonradial $g$-mode pulsations are triggered by hydrogen burning, with the aim of placing constraints on hydrogen shell burning in cool white dwarfs and on a third dredge-up during the asymptotic giant branch evolution of their progenitor stars. We construct white-dwarf sequences from low-metallicity progenitors by means of full evolutionary calculations, and analyze their pulsation stability for the models in the range of effective temperatures $T_{rm eff} sim 15,000,-, 8,000$ K. We demonstrate that, for white dwarf models with masses $M_{star} lesssim 0.71,rm M_{sun}$ and effective temperatures $8,500 lesssim T_{rm eff} lesssim 11,600$ K that evolved from low-metallicity progenitors ($Z= 0.0001$, $0.0005$, and $0.001$) the dipole ($ell= 1$) and quadrupole ($ell=2$) $g_1$ modes are excited mostly due to the hydrogen-burning shell through the $varepsilon$-mechanism, in addition to other $g$ modes driven by either the $kappa-gamma$ or the convective driving mechanism. However, the $varepsilon$ mechanism is insufficient to drive these modes in white dwarfs evolved from solar-metallicity progenitors. We suggest that efforts should be made to observe the dipole $g_1$ mode in white dwarfs associated with low-metallicity environments, such as globular clusters and/or the galactic halo, to place constraints on hydrogen shell burning in cool white dwarfs and the third dredge-up episode during the preceding asymptotic giant branch phase.
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