Alzheimer's disease (AD) is a leading cause of disability worldwide. Early detection is critical for preventing progression and formulating effective treatment plans. This study aims to develop a novel deep learning (DL) model, Hybrid-RViT, to enhance the detection of AD. The proposed Hybrid-RViT model integrates the pre-trained convolutional neural network (ResNet-50) with the Vision Transformer (ViT) to classify brain MRI images across different stages of AD. The ResNet-50 adopted for transfer learning, facilitates inductive bias and feature extraction. Concurrently, ViT processes sequences of image patches to capture long-distance relationships via a self-attention mechanism, thereby functioning as a joint local-global feature extractor. The Hybrid-RViT model achieved a training accuracy of 97% and a testing accuracy of 95%, outperforming previous models. This demonstrates its potential efficacy in accurately identifying and classifying AD stages from brain MRI data. The Hybrid-RViT model, combining ResNet-50 and ViT, shows superior performance in AD detection, highlighting its potential as a valuable tool for medical professionals in interpreting and analyzing brain MRI images. This model could significantly improve early diagnosis and intervention strategies for AD.