Task offloading in Vehicular Edge Computing (VEC) can advance cooperative perception (CP) to improve traffic awareness in Autonomous Vehicles. In this paper, we propose the Quality-aware Cooperative Perception Task Offloading (QCPTO) scheme. Q-CPTO is the first task offloading scheme that enhances traffic awareness by prioritizing the quality rather than the quantity of cooperative perception. Q-CPTO improves the quality of CP by curtailing perception redundancy and increasing the Value of Information (VOI) procured by each user. We use Kalman filters (KFs) for VOI assessment, predicting the next movement of each vehicle to estimate its region of interest. The estimated VOI is then integrated into the task offloading problem. We formulate the task offloading problem as an Integer Linear Program (ILP) that maximizes the VOI of users and reduces perception redundancy by leveraging the spatially diverse fields of view (FOVs) of vehicles, while adhering to strict latency requirements. We also propose the Q-CPTO-Heuristic (Q-CPTOH) scheme to solve the task offloading problem in a time-efficient manner. Extensive evaluations show that Q-CPTO significantly outperforms prominent task offloading schemes by up to 14% and 20% in terms of response delay and traffic awareness, respectively. Furthermore, Q-CPTO-H closely approaches the optimal solution, with marginal gaps of up to 1.4% and 2.1% in terms of traffic awareness and the number of collaborating users, respectively, while reducing the runtime by up to 84%.