Hydrogen incorporation into magnetic materials offers a powerful way to modulate spintronics functionalities through changes in electronic states and magnetic interactions. We show that the hydrogen-induced modulation of the magnetization compensation temperature Tcomp is strongly dependent on nonmagnetic underlayer materials. Ferrimagnetic Co/Tb systems with various underlayers were used in this study. The sign of anomalous Hall effect was reversed after H2 gas exposure in the Pt underlayer samples, whereas no change in sign was observed in the Cu underlayer sample. We also found that H2 exposure caused a substantial decrease in Tcomp in the Pt and Pd underlayer samples, suggesting that the strong catalytic effect of Pt and Pd promotes hydrogen incorporation into the Co/Tb systems. Magnetization measurements and calculations based on molecular field approximation indicated that the weakening of the antiferromagnetic Co-Tb exchange coupling is a plausible mechanism of the Tcomp reduction. Our study highlights that in ferrimagnet/nonmagnet layered systems, nonmagnetic underlayer is crucial not only for the magnetic properties but also for the response to hydrogen.