Accurate vehicle trajectory prediction is critical for autonomous vehicles and advanced driver assistance systems to make driving decisions and improve traffic safety. This paper proposes a novel Temporal Multi-task Mixture Of Experts (TMMOE) model for simultaneously predicting the vehicle trajectory and driving intention, considering interconnections among tasks. As for the methodology, the proposed model consists of three layers: a shared layer, an expert layer, and a fully connected layer. In more detail, the first layer utilizes Temporal Convolutional Network (TCN) to extract temporal features whereas the expert layer incorporates the gating mechanism to memorize and filter the temporal dependence of sequences and, finally, the fully connected layer is applied to integrate and export prediction results. Furthermore, the homoscedastic uncertainty algorithm is used to construct the multi-task loss function. The open data source CitySim dataset is chosen to validate the performance of the proposed TMMOE model; moreover, a novel lane line reconstruction method is introduced to mitigate measurement errors of the dataset. Two-part information, including the history of the vehicle's trajectory and the interaction indicators, are employed as the input variables. The result indicates that the TMMOE algorithm exhibits superior performance when compared to the other five models, including Long Short-Term Memory (LSTM) Network, Convolutional Neural Network (CNN), CNN-LSTM, TCN, and Gate Recurrent Unit (GRU), while considering varying input sequence lengths are 3 s, 6 s, and 9 s respectively. Finally, the sensitivity analysis demonstrates that considering driving intentions in vehicle trajectory prediction can significantly enhance the prediction accuracy of vehicle trajectories.