The accurate prediction of vehicle state based on the data acquired by the vehicular networking system plays an important role in improving traffic safety in the transportation section. However, it is difficult to accurately predict the vehicle state due to the highly dynamic road environment and various drivers' behaviors. To this end, in this article, we propose a two-level quantized adaptive Kalman filter (KF) algorithm based on the autoregressive moving average (MA) model to predict the vehicle state (including the moving direction, driving lane, vehicle speed, and acceleration). First, we propose a vehicular networking system to acquire the vehicle data by exchanging traffic data between the onboard unit and the roadside unit (RSU). Then, we predict the vehicle state at the edge cloud server (ECS) equipped at the RSU. Specifically, we utilize the autoregressive MA model to predict vehicle acceleration at the next moment. Then, the predicted vehicle acceleration is used as an input variable of the adaptive KF model to predict the vehicle location and speed at the next moment, in which we quantify the predicted vehicle location to the moving direction and the driving lane. Finally, the ECS broadcasts the predicted state to other RSUs. Through the communication with the road unit, all vehicles moving at the intersection can share vehicles states each other. In this doing, we can efficiently improve traffic safety in the intersection. We provide numerical simulations to validate the effectiveness of the autoregressive MA model used for predicting acceleration. Then, we evaluate the efficiency of the proposed two-level quantized adaptive KF algorithm. Compared with five conventional prediction algorithms, our proposed algorithm can improve the speed prediction accuracy by 90.62%, 89.81%, 88.91%, 82.76%, and 70.77%, respectively, which implies that our algorithm is a promising scheme for predicting the vehicle state in vehicular networks.