Multiphoton photoemission of laser irradiated alpha -SiO2 is studied as a function of different laser parameters. The laser is a frequency-doubled or -tripled Nd-YAG laser, operated in the picosecond pulse duration regime. The sample is a single crystal, cut perpendicularly to the C-axis direction, kept under ultra-high vacuum (UHV) conditions. The intensity dependence of the photocurrent reveals that emission at the second harmonic frequency is throughout the whole accessible intensity range governed by defect states, whereas at the third harmonic frequency the authors observe a transition from a two- to three-photon dynamics, which could reveal the onset of the valence band emission. However, the most probable explanation is still in both cases a two-photon defect-driven process, eventually followed by electron heating in the conduction band through electron-photon-phonon collisions, or by photon absorption by electrons trapped in the bottom of the conduction band. The dependence of the photocurrent from the laser polarization direction is shown to differ depending on whether all the emitted electrons are collected or only the ones emitted along the C-axis. In this last case, a selection rule seems to apply to the emission process that yields information about the symmetry of the emitting state.