We conduct an analysis to classify sleep stages using electroencephalogram (EEG) data from mice, focusing on Quantile Graphs. The method constructs networks from EEG time series data and uses the extracted network features to classify sleep stages through a random forest. A key aspect of the proposed approach is the inclusion of network-based features such as clustering coefficients and average shortest path length, in addition to traditional statistical features. Since the Quantile Graph is derived from network data, it produces distinct results that differ from those obtained using traditional features. The experiments demonstrate a notable improvement in classification accuracy for rapid eye movement (REM), while the network features are less effective for other stages, such as non-REM (NREM), high frequency theta wave (HT), and low frequency theta wave (LT). The results indicate that the choice of time delay and quantization settings plays a crucial role in achieving accurate classification of sleep stages, with the optimal time delay for maximum classification performance for REM identified at approximately 83.3 ms.