Automated Guided Vehicles (AGVs) are essential in various industries for their efficiency and adaptability. However, planning trajectories for AGVs in obstacle-dense, unstructured environments presents significant challenges due to the nonholonomic kinematics, abundant obstacles, and the scenario's nonconvex and constrained nature. To address this, we propose an efficient trajectory planning framework for AGVs by formulating the problem as an optimal control problem. Our framework utilizes the fast safe rectangular corridor (FSRC) algorithm to construct rectangular convex corridors, representing avoidance constraints as box constraints. This eliminates redundant obstacle influences and accelerates the solution speed. Additionally, we employ the Modified Visibility Graph algorithm to speed up path planning and a boundary discretization strategy to expedite FSRC construction. Experimental results demonstrate the effectiveness and superiority of our framework, particularly in computational efficiency. Compared to advanced frameworks, our framework achieves computational efficiency gains of 1 to 2 orders of magnitude. Notably, FSRC significantly outperforms other safe convex corridor-based methods regarding computational efficiency.