Conventional motion estimation techniques in ultrasound images such as exhaustive search-based block matching (ES-BM) have been studied exhaustively and are known to be computationally expensive and slow. Consequently, they are not feasible for real-time processing. On the other hand, several deep learning-based techniques are being developed for realtime motion estimation of day-to-day objects. In this paper, we attempt to bridge the gap between tracking techniques being used for ultrasound images and recent deep learning-based techniques used for non-medical real-world objects. We propose to adopt the deep neural network-based Fully-Convolutional Siamese tracker (SiamFC) to track regions of interest (ROI) in ultrasound images. We prove that siamese architecture-based tracker is feasible for motion tracking in ultrasound images and performs better than conventional ES-BM technique. We applied SiamFC and ES-BM on 10 different image sequences to track the motion of the transverse section of the carotid artery. Our experiments showed that SiamFC was almost six times faster with slightly better performance compared to ES-BM in most of the cases.