We report the discovery of the disappearance of Mg II, Al III, C IV, and Si IV broad absorption lines (BALs) at the same velocity (0.07c), accompanied by a new C IV BAL emerging at a higher velocity (up to 0.11c), in the quasar J0827+4252 at z = 2.038. This is the first report of BAL disappearance (i) over Mg II, Al III, C IV, and Si IV ions and (ii) in a weak emission-line quasar (WLQ). The discovery is based on four spectra from the Sloan Digital Sky Survey and one follow-up spectrum from Hobby-Eberly Telescope/Low-Resolution Spectrograph-2. The simultaneous C IV BAL disappearance and emergence at different velocities, together with no variations in the Catalina Real-Time Transient Survey light curve, indicate that ionization changes in the absorbing material are unlikely to cause the observed BAL variability. Our analyses reveal that transverse motion is the most likely dominant driver of the BAL disappearance/emergence. Given the presence of mildly relativistic BAL outflows and an apparently large C IV emission-line blueshift that is likely associated with strong bulk outflows in this WLQ, J0827+4252 provides a notable opportunity to study extreme quasar winds and their potential in expelling material from inner to large-scale regions.