Monitoring sleep of premature infants is a vital aspect of clinical care, as it can reveal potential future pathologies and health issues. This study presents a novel approach to automatically estimate and track Quiet Sleep (QS) in 33 newborns using ECG, respiration, and video motion features. Using an annotated dataset from 15 neonates (10 preterm, 5 full-term) encompassing 127.2 hours, a comprehensive feature extraction and selection process was employed. Three classifiers (Random Forest, Logistic Regression, K-Nearest Neighbors) were evaluated to develop a QS estimation model. A compact and interpretable model was selected, achieving a balanced accuracy of 84.67.5%. The robustness of the model was further enhanced by incorporating a switching mechanism between models using only ECG and respiration when video data was unavailable. The study further explored the evolution of QS during hospitalization using a large dataset with 18 newborns (16 preterm and 2 term) and 1396.6 hours of data. It highlighted an increase in QS duration and mean interval duration with post-menstrual age. The results offer valuable insights into the developmental progress of healthy preterm infants and underscore the potential of continuous, non-invasive monitoring in neonatal intensive care units.