Motivated by the inconceivable capability of the human brain in simultaneously processing multi-modal signals and its real-time feedback to the outer world events, there has been a surge of interest in establishing a communication bridge between the human brain and a computer, which are referred to as Brain-computer Interfaces (BCI). To this aim, monitoring the electrical activity of brain through Electroencephalogram (EEG) has emerged as the prime choice for BCI systems. To discover the underlying and specific features of brain signals for different mental tasks, a considerable number of research works are developed based on statistical and data-driven techniques. However, a major bottleneck in the development of practical and commercial BCI systems is their limited performance when the number of mental tasks for classification is increased. In this work, we propose a new EEG processing and feature extraction paradigm based on Siamese neural networks, which can be conveniently merged and scaled up for multi-class problems. The idea of Siamese networks is to train a double-input neural network based on a contrastive loss-function, which provides the capability of verifying if two input EEG trials are from the same class or not. In this work, a Siamese architecture, which is developed based on Convolutional Neural Networks (CNN) and provides a binary output on the similarity of two inputs, is combined with OVR and OVO techniques to scale up for multi-class problems. The efficacy of this architecture is evaluated on a 4-class Motor Imagery (MI) dataset from BCI Competition IV-2a and the results suggest a promising performance compared to its counterparts.