Traffic flow forecasting on graphs has real-world applications in many fields, such as transportation system and computer networks. Traffic forecasting can be highly challenging due to complex spatial-temporal correlations and non-linear traffic patterns. Existing works mostly model such spatial-temporal dependencies by considering spatial correlations and temporal correlations separately and fail to model the direct spatial-temporal correlations. Inspired by the recent success of transformers in the graph domain, in this paper, we propose to directly model the cross-spatial-temporal correlations on the spatial-temporal graph using local multi-head self-attentions. To reduce the time complexity, we set the attention receptive field to the spatially neighboring nodes, and we also introduce an adaptive graph to capture the hidden spatial-temporal dependencies. Based on these attention mechanisms, we propose a novel Adaptive Graph Spatial-Temporal Transformer Network (ASTTN), which stacks multiple spatial-temporal attention layers to apply self-attention on the input graph, followed by linear layers for predictions. Experimental results on public traffic network datasets, METR-LA PEMS-BAY, PeMSD4, and PeMSD7, demonstrate the superior performance of our model.