Hydrogen contamination in Josephson junctions may contribute to device-to-device variability in superconducting qubits, yet its incorporation, location, and bonding in tunnel barriers remain poorly understood. We combine molecular dynamics simulations of aluminum oxidation in dense O2 and H2O environments with atomistic quantum transport calculations. The simulations produce amorphous AlOx barriers with experimentally relevant hydrogen concentrations. Analysis of 400 statistically independent samples shows that the number of incorporated H atoms follows a beta-binomial distribution, reflecting correlations arising from self-limiting oxidation. Most H atoms reside near the AlOx surface, primarily in Al-OH and Al-OH-Al motifs. Quantum transport calculations for Al/Al2O3/Al junctions reveal that hydrogen enhances transmission near the Fermi level by creating an interface dipole that shifts the oxide bands relative to the electrode Fermi level. Combining hydrogen-number statistics with calculated transmission coefficients yields a probability distribution for the Josephson energy. At an average hydrogen content of 2.56 at.%, we predict E_J/h = 10.23 GHz with a standard deviation of 3.6 MHz. Hydrogen shifts the mean Josephson energy by approximately 11%, proportionally to its concentration, but contributes far less to device-to-device variability than observed in state-of-the-art fabricated junctions. These findings provide an atomistic understanding of hydrogen incorporation and its impact on Josephson energy.