(1) Background: Ultrasound provides a radiation-free and portable method for assessing swallowing. Hyoid bone locations and displacements are often used as important indicators for the evaluation of swallowing disorders. However, this requires clinicians to spend a great deal of time reviewing the ultrasound images. (2) Methods: In this study, we applied tracking algorithms based on deep learning and correlation filters to detect hyoid locations in ultrasound videos collected during swallowing. Fifty videos were collected from 10 young, healthy subjects for training, evaluation, and testing of the trackers. (3) Results: The best performing deep learning algorithm, Fully-Convolutional Siamese Networks (SiamFC), proved to have reliable performance in getting accurate hyoid bone locations from each frame of the swallowing ultrasound videos. While having a real-time frame rate (175 fps) when running on an RTX 2060, SiamFC also achieved a precision of 98.9% at the threshold of 10 pixels (3.25 mm) and 80.5% at the threshold of 5 pixels (1.63 mm). The tracker's root-mean-square error and average error were 3.9 pixels (1.27 mm) and 3.3 pixels (1.07 mm), respectively. (4) Conclusions: Our results pave the way for real-time automatic tracking of the hyoid bone in ultrasound videos for swallowing assessment.