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Sleep plays an important role in neonatal brain and physical development, making its detection and characterization important for assessing early-stage development. In this study, we propose an automatic and computationally efficient algorithm to detect neonatal quiet sleep (QS) using a convolutional neural network (CNN). Our study used 38-hours of electroencephalography (EEG) recordings, collected from 19 neonates at Fudan Children's Hospital in Shanghai, China (Approval No. (2020) 22). To train and test the CNN, we extracted 12 prominent time and frequency domain features from 9 bipolar EEG channels. The CNN architecture comprised two convolutional layers with pooling and rectified linear unit (ReLU) activation. Additionally, a smoothing filter was applied to hold the sleep stage for 3 minutes. Through performance testing, our proposed method achieved impressive results, with 94.07% accuracy, 89.70% sensitivity, 94.40% specificity, 79.82% F1-score and a 0.74 kappa coefficient when compared to human expert annotations. A notable advantage of our approach is its computational efficiency, with the entire training and testing process requiring only 7.97 seconds. The proposed algorithm has been validated using leave one subject out (LOSO) validation, which demonstrates its consistent performance across a diverse range of neonates. Our findings highlight the potential of our algorithm for real-time neonatal sleep stage classification, offering a fast and cost-effective solution. This research opens avenues for further investigations in early-stage development monitoring and the assessment of neonatal health.