Travel time reliability (TTR) serves as a crucial indicator for evaluating the efficiency and service quality of a road traffic network. This paper proposes a multi-task spatio-temporal generative adversarial network (MTST-GAN) model that simultaneously predicts the TTR in morning and evening peak hour periods. The model incorporates multi-graph convolutional networks to extract spatial correlations from travel time data, while long short-term memory neural networks are employed to consider temporal correlations. Additionally, self-attention mechanisms are applied to the proposed MTST-GAN model to further capture spatial and temporal features. A feature fusion bridge is constructed to integrate the spatial and temporal features learned by each task. Through a numerical experiment conducted on a road network in a Chinese city, our findings demonstrate that the proposed model outperforms several state-of-the-art approaches in terms of Jensen-Shannon divergence, mean, standard deviation, and buffer time indices. Finally, we provide conclusions and suggest areas for further research.