The increasing demand for aquaculture production necessitates the development of innovative, intelligent tools to effectively monitor and manage fish health and welfare. While non-invasive video monitoring has become a common practice in finfish aquaculture, existing intelligent monitoring methods predominantly focus on assessing body condition or fish swimming patterns and are often developed and evaluated in controlled tank environments, without demonstrating their applicability to real-world aquaculture settings in open sea farms. This underscores the necessity for methods that can monitor physiological traits directly within the production environment of sea fish farms. To this end, we have developed a computer vision method for monitoring ventilation rates of Atlantic salmon (Salmo salar), which was specifically designed for videos recorded in the production environment of commercial sea fish farms using the existing infrastructure. Our approach uses a fish head detection model, which classifies the mouth state as either open or closed using a convolutional neural network. This is followed with multiple object tracking to create temporal sequences of fish swimming across the field of view of the underwater video camera to estimate ventilation rates. The method demonstrated high efficiency, achieving a Pearson correlation coefficient of 0.82 between ground truth and predicted ventilation rates in a test set of 100 fish collected independently of the training data. By accurately identifying pens where fish exhibit signs of respiratory distress, our method offers broad applicability and the potential to transform fish health and welfare monitoring in finfish aquaculture.