The functional connectivity (FC) between brain regions is usually estimated through a statistical dependency method with functional magnetic resonance imaging (fMRI) data. It inevitably yields redundant and noise connections, limiting the performance of deep supervised models in brain disease diagnosis. Besides, the supervised signals of fMRI data are insufficient due to the shortage of labeled data. To address these issues, we propose an end-to-end unsupervised graph structure learning method for sufficiently capturing the structure or characteristics of the functional brain network itself without relying on manual labels. More specifically, the proposed method incorporates a graph generation module for automatically learning the discriminative graph structures of functional brain networks and a topology-aware encoding module for sufficiently capturing the structure information. Furthermore, we also design view consistency and correlation-guided contrastive regularizations. We evaluated our model on two real medical clinical applications: the diagnosis of Bipolar Disorder (BD) and Major Depressive Disorder (MDD). The results suggest that the proposed method outperforms state-of-the-art methods. In addition, our model is capable of identifying associated biomarkers and providing evidence of disease association. To the best of our knowledge, our work attempts to construct learnable functional brain networks with unsupervised graph structure learning. Our code is available at https://github.com/IntelliDAL/Graph/tree/main/BrainUSL.