Alzheimer’s Disease (AD) is an irreversible neurodegenerative disease, the most common form of dementia, affecting millions worldwide. Neuroimaging-based early AD diagnosis has become an effective approach, especially by using structural Magnetic Resonance Imaging (sMRI). The convolutional neural network (CNN) based method is challenging to learn dependencies between spatially distant positions in the various brain regions due to its local convolution operation. In contrast, the graph convolutional network (GCN) based work can connect the brain regions to capture global information but is not sensitive to the local information in a single brain region. Unlike a separate CNN or GCN-based method, we proposed a brain-inspired global-local information fusion network (BGL-Net) to diagnose AD. It essentially inherits the advantages of both CNN and GCN. The experiments on three public datasets demonstrate the effectiveness and robustness of our BGL-Net. Our method achieved the best performance on three popular public datasets compared with the existing CNN and GCN-based methods. In addition, our visualization results of the learned brain connection on AD and normal people agree with many current AD clinical research.