Autism spectrum disorder (ASD) is a range of neurodevelopmental disorders, which brings enormous burdens to the families of patients and society. However, due to the lack of representation of variance for diseases and the absence of biomarkers for diagnosis, the early detection and intervention of ASD are remarkably challenging. In this study, we proposed a self-attention deep learning framework based on the transformer model on structural MR images from the ABIDE consortium to classify ASD patients from normal controls and simultaneously identify the structural biomarkers. In our work, the individual structural covariance networks are used to perform ASD/NC classification via a self-attention deep learning framework, instead of the original structural MR data, to take full advantage of the coordination patterns of morphological features between brain regions. The self-attention deep learning framework based on the transformer model can extract both local and global information from the input data, making it more suitable for the brain network data than the CNN- structural model. Meanwhile, the potential diagnosis structural biomarkers are identified by the self-attention coefficients map. The experimental results showed that our proposed method outperforms most of the current methods for classifying ASD patients with the ABIDE data and achieves a classification accuracy of 72.5% across different sites. Furthermore, the potential diagnosis biomarkers were found mainly located in the prefrontal cortex, temporal cortex, and cerebellum, which may be treated as the early biomarkers for the ASD diagnosis. Our study demonstrated that the self-attention deep learning framework is an effective way to diagnose ASD and establish the potential biomarkers for ASD.