Using first-principles density functional theory calculations, we have investigated the O vacancy formation and the relevant induced defect states in hafnium silicates over a wide range of compositions. The PBE0 hybrid density functional was employed for the analysis of the electronic properties and the charge transition levels of the O vacancy in crystalline HfSiO4 and in amorphous Hf-silicates, respectively. Based on the generated structure models, eight typical kinds of O coordination structures were identified in amorphous Hf-silicates. Our calculated results show that the positions of the induced defect energy levels in the band gap and the formation energies of O vacancy are largely determined by the local structures of the vacancy sites, which appear to be nearly independent of the composition of amorphous Hf-silicates. Our calculations also show that O vacancy can possess the negative-U behavior in crystalline HfSiO4 but not in amorphous Hf-silicates, where most of the O vacancies can simply exhibit the negative-U behavior as in the positive charge states. Given the measured band offset of 3.40 eV between Si and amorphous Hf-silicates, a considerable number of O vacancies were found to prefer to stay in the charge neutral state as the Fermi level lies within the band gap region of Si. Furthermore, due to its relatively higher formation energy, the concentration of O vacancy in Hf-silicates can be much lower than that in m-HfO2 when the Fermi level lies below the midgap region of Si. Accordingly, a significantly reduced flat band voltage shift and less transient threshold voltage instability can be found in Hf-silicates as compared with m-HfO2, which are in good agreement with the recent experimental findings.