Alzheimer's disease (AD) may cause damage to the memory cells permanently, which results in the form of dementia. The diagnosis of Alzheimer's disease at an early stage is a problematic task for researchers. For this, machine learning and deep convolutional neural network (CNN) based approaches are readily available to solve various problems related to brain image data analysis. In clinical research, magnetic resonance imaging (MRI) is used to diagnose AD. For accurate classification of dementia stages, we need highly discriminative features obtained from MRI images. Recently advanced deep CNN-based models successfully proved their accuracy. However, due to a smaller number of image samples available in the datasets, there exist problems of over-fitting hindering the performance of deep learning approaches. In this research, we developed a Siamese convolutional neural network (SCNN) model inspired by VGG-16 (also called Oxford Net) to classify dementia stages. In our approach, we extend the insufficient and imbalanced data by using augmentation approaches. Experiments are performed on a publicly available dataset open access series of imaging studies (OASIS), by using the proposed approach, an excellent test accuracy of 99.05% is achieved for the classification of dementia stages. We compared our model with the state-of-the-art models and discovered that the proposed model outperformed the state-of-the-art models in terms of performance, efficiency, and accuracy.