In this study, the adsorption of hydrogen and methane on the Zn2(NDC)2(diPyTz) [(NDC = 2,6-naphthalenedicarboxylate, diPyTz = di-3,6-(4-pyridyl)-1,2,4,5-tetrazine)] metal–organic framework (MOF) and the effect of its doping with alkali metal cations (Li+, Na+, K+) were investigated using Grand Canonical Monte Carlo simulations. The results indicated that the triply catenating Zn2(NDC)2(diPyTz), possessing small pores preferentially adsorbed hydrogen. Doping of Zn2(NDC)2(diPyTz) with alkali metal cations enhanced the hydrogen adsorption on the MOF. However, this enhancement became weaker as the atomic number of metal cation increased. The simulation results showed that the hydrogen adsorption on the Li+-doped Zn2(NDC)2(diPyTz) was almost 2.35 times greater than that of the corresponding undoped MOF at low pressure and room temperature. This suggests that the doping of MOFs with alkali metal cations especially lithium is a desired strategy for hydrogen storage. Furthermore, the results revealed that the adsorption of hydrogen on the Zn2(NDC)2(diPyTz) was higher than that of methane at room temperature.