In vehicular networks, RoadSide Units (RSUs) are not available everywhere, therefore, it is not possible for vehicles to stay connected with RSUs all the time and to send their data directly to RSUs anytime using Vehicle-to-Infrastructure (V2I) data offloading. Hence, in these cases, Vehicle-to-Vehicle (V2V) communications are used to offload data to RSUs through other vehicles. Data to offload could be urgent (e.g., accident data), therefore, it is important to consider the Quality of Service (QoS) provisioning. In this paper, we propose a QoS-aware data offloading scheme for vehicular networks with QoS provisioning (DOVEQ) that considers both V2I and V2V data offloading. DOVEQ models the connectivity of vehicles with RSUs and vehicles, offloading capacity and estimation of reaching RSUs. It provides QoS using traffic classification, overload control and admission control. The performance evaluation of DOVEQ shows that DOVEQ outperforms other schemes by offloading more amount of important data with lesser offloading delay and running time.