hydrogen incorporation in oxidized aluminum barriers by combining molecular dynamics simulations with atomistic quantum transport calculations. The oxide growth simulations are performed using CHGNet for Al surfaces exposed to dense O2 and H$_{\text{2}% }$O environments, yielding amorphous AlOx layers with hydrogen content comparable to experimentally relevant levels. From 400 statistically independent samples, we find that the number of H atoms in the oxide is well described by a beta-binomial distribution, reflecting correlations induced by the self-limiting oxidation process. Structural analysis shows that most hydrogen atoms reside near the AlOx surface and predominantly form Al-OH and Al-OH-Al motifs. To assess the impact of hydrogen on transport, we construct Al/Al2O3/Al junction models and perform NEGF-DFT calculations with NanoDCAL, using a GGA+U scheme to calibrate the band gap and band alignment. H atoms are found to increase the transmission coefficient near the Fermi level and shift the electronic structure in a manner consistent with effective p-type doping. By combining the H atom number statistics from molecular dynamics with the transmission coefficients from quantum transport calculations, we obtain a probability distribution for the Josephson energy. For a Josephson junction with an average hydrogen content of 2.56 at.\%, the resulting Josephson energy is predicted to be GHz. These results provide an atomistic picture of hydrogen contamination and an estimate of device variability in Josephson junctions.