Alzheimer's disease (AD) is a neurodegenerative disease that causes irreversible damage to several brain regions, including the hippocampus causing impairment in cognition, function, and behaviour. Early diagnosis of the disease will reduce the suffering of the patients and their family members. Towards this aim, in this paper, we propose a Siamese Convolutional Neural Network (CNN) architecture that employs the triplet-loss function for the representation of input MRI images as k-dimensional embeddings. We used both pre-trained and non-pre-trained CNNs to transform images into the embedding space. These embeddings are subsequently used for the 4-way classification of Alzheimer's Disease. The model efficacy was tested using the ADNI and OASIS datasets which produced an accuracy of 91.83% and 93.85%, respectively. Furthermore, obtained results are compared with similar methods proposed in the literature.