Local configurations and localized vibrations of oxygen interstitials in different charged states in alpha-quartz are investigated by computer calculations. First-principle potentials of the Buckingham type have been used in numerical modeling. The study of the lattice dynamics in defective crystal is performed using the phonon local density of states. Frequencies of localized vibrations induced by oxygen interstitials are determined. It is shown that the atomic configuration, type, number and frequency of localized vibrations depend on the sign and magnitude of the interstitial atom charge.