Alzheimer’s disease (AD) is a progressive and irreversible neurodegenerative disease. Neuroimaging based on magnetic resonance imaging (MRI) is one of the most intuitive and reliable methods to perform AD screening and diagnosis. Clinical head MRI detection generates multimodal image data, and to solve the problem of multimodal MRI processing and information fusion, this paper proposes a structural and functional MRI feature extraction and fusion method based on generalized convolutional neural networks (gCNN). The method includes a three-dimensional residual U-shaped network based on hybrid attention mechanism (3D HA-ResUNet) for feature representation and classification for structural MRI, and a U-shaped graph convolutional neural network (U-GCN) for node feature representation and classification of brain functional networks for functional MRI. Based on the fusion of the two types of image features, the optimal feature subset is selected based on discrete binary particle swarm optimization, and the prediction results are output by a machine learning classifier. The validation results of multimodal dataset from the AD Neuroimaging Initiative (ADNI) open-source database show that the proposed models have superior performance in their respective data domains. The gCNN framework combines the advantages of these two models and further improves the performance of the methods using single-modal MRI, improving the classification accuracy and sensitivity by 5.56% and 11.11%, respectively. In conclusion, the gCNN-based multimodal MRI classification method proposed in this paper can provide a technical basis for the auxiliary diagnosis of Alzheimer’s disease.