Automated sleep stage classification is imperative for detecting sleep-related disorders. Previous studies predominantly favored single-channel sleep signals for their computational efficiency. However, the present research endeavor advances a novel approach, Randomized Quaternion Minimal Gated Unit (RQMGU), for multichannel sleep stage classification. RQMGU integrates Minimal Gated Unit, a simplified variant of traditional Recurrent Neural Networks, and employs quaternions to capture internal channel dependencies. Additionally, Random Projection is seamlessly integrated as a data representation mechanism, optimizing efficiency-performance trade-offs without employing dimensionality reduction. Despite incorporating multiple channels, RQMGU maintains a parsimonious architecture, achieving up to a substantial 52-fold reduction in training parameters as opposed to compared models, resulting in significantly lower computational resource requirements. Empirical findings on the Sleep-EDF-78 dataset underscore the efficacy of RQMGU, demonstrating comparable accuracy to contemporary baseline methods.