In this paper, we present a trajectory planning method to navigate a quadrotor with a cable-suspended payload through known obstacle-filled environments. We model the system as a hybrid dynamical system and formulate the trajectory generation problem as a Mixed Integer Quadratic Program (MIQP). Specifically, we address two novel challenges. First, we plan for a multi-body system, and obstacle avoidance must be guaranteed for the quadrotor, load, and the cable. Second, our method accommodates transitions between subsystems of the hybrid dynamical system, allowing for maneuvers that would otherwise be infeasible if the cable were constrained to remain taut. Numerical and experimental results validate the proposed approach for the full hybrid system.