Respiratory sound is able to differentiate sleep stages and provide a non-contact and cost-effective solution for the diagnosis and treatment monitoring of sleep-related diseases. While most of the existing respiratory sound-based methods focus on a limited number of sleep stages such as sleep/wake and wake/rapid eye movement (REM)/non-REM, it is essential to detect sleep stages at a finer level for sleep quality evaluation. In this paper, we for the first time study a sleep stage detection method aiming at classifying sleep states into four sleep stages: wake, REM, light sleep, and deep sleep from the respiratory sound. In addition to extracting time-domain features, frequency-domain features of respiratory sound, non-linear features of snoring sound are devised to better characterize snoring-related signals of respiratory sound. To effectively fuse the three sets of features, a novel feature fusion technique combining the generalized canonical correlation analysis with the ReliefF algorithm is proposed for discriminative feature selection. Final stage detection is achieved with popular classifiers including decision tree, support vector machines, K-nearest neighbor, and the ensemble classifier. To evaluate our proposed method, we built an in-house dataset, which is comprised of 13 nights of sleep audio data from a sleep laboratory. Experimental results indicate that our proposed method outperforms the existing related ones and is promising for large-scale non-contact sleep monitoring.