This paper focuses on the safe navigation of quadrotor unmanned aerial vehicles (UAVs) under obstacle-related safety constraints and prescribed tracking performance within a user-assigned predefined time. A novel control strategy is developed by integrating a Barrier Lyapunov Function (BLF) with an error-shifting transformation into the translational control mechanism, ensuring the vehicle stays clear of a safety sphere constructed around potential obstacles while being guided to a target position. This approach effectively handles arbitrary initial conditions and maintains safe operation within predefined bounds. Furthermore, by combining a prescribed-performance mechanism with a time-varying performance function and the backstepping technique, the proposed method allows the designer to specify in advance the performance-bound attainment time $T_0$ and the terminal tracking-error bound $s_{T_0}$. For the analyzed translational closed-loop subsystem, all signals are bounded, and the tracking error is theoretically shown to enter the set $|e_{1j}|<s_{T_0}$ no later than $T_0$. With the designed controller, all signals in the analyzed translational closed-loop subsystem are bounded, and the output tracking error is theoretically shown to enter a user-defined prescribed set no later than the performance-bound attainment time $T_0$. Simulation studies demonstrate the effectiveness of the designed controller in achieving desired performance metrics.