We perform a detailed quantum dynamical study of nonequilibrium Josephson oscillations between interacting Bose-Einstein condensates confined in a finite-size double-well trap. We find that the Josephson junction can sustain multiple undamped Josephson oscillations up to a characteristic time scale τc without quasipartcles being excited in the system. This may explain recent related experiments. Beyond a characteristic time scale τc the dynamics of the junction is governed by fast, quasiparticle-assisted Josephson tunneling as well as Rabi oscillations between the discrete quasiparticle levels. We predict that an initially self-trapped state of the Bose-Einstein condensates will be destroyed by these fast dynamics.