Rapid advances in perception have enabled large pre-trained models to be used out of the box for transforming high-dimensional, noisy, and partial observations of the world into rich occupancy representations. However, the reliability of these models and consequently their safe integration onto robots remains unknown when deployed in environments unseen during training. To provide safety guarantees, we rigorously quantify the uncertainty of pre-trained perception systems for object detection and scene completion via a novel calibration technique based on conformal prediction. Crucially, this procedure guarantees robustness to distribution shifts in states when perception outputs are used in conjunction with a planner. As a result, the calibrated perception system can be used in combination with any safe planner to provide an end-to-end statistical assurance on safety in unseen environments. We evaluate the resulting approach, Perceive with Confidence (PwC), in simulation and on hardware where a quadruped robot navigates through previously unseen indoor, static environments. These experiments validate the safety assurances for obstacle avoidance provided by PwC. In simulation, our method reduces obstacle misdetection by $70\%$ compared to uncalibrated perception models. While misdetections lead to collisions for baseline methods, our approach consistently achieves $100\%$ safety. We further demonstrate reducing the conservatism of our method without sacrificing safety, achieving a $46\%$ increase in success rates in challenging environments while maintaining $100\%$ safety. In hardware experiments, our method improves empirical safety by $40\%$ over baselines and reduces obstacle misdetection by $93.3\%$. The safety gap widens to $46.7\%$ when navigation speed increases, highlighting our approach's robustness under more demanding conditions.