Automatic sleep stage classification plays an essential role in sleep quality measurement and sleep disorder diagnosis. Although many approaches have been developed, most use only single-channel electroencephalogram signals for classification. Polysomnography (PSG) provides multiple channels of signal recording, enabling the use of the appropriate method to extract and integrate the information from different channels to achieve higher sleep staging performance. We present a transformer encoder-based model, MultiChannelSleepNet, for automatic sleep stage classification with multichannel PSG data, whose architecture is implemented based on the transformer encoder for single-channel feature extraction and multichannel feature fusion. In a single-channel feature extraction block, transformer encoders extract features from time-frequency images of each channel independently. Based on our integration strategy, the feature maps extracted from each channel are fused in the multichannel feature fusion block. Another set of transformer encoders further capture joint features, and a residual connection preserves the original information from each channel in this block. Experimental results on three publicly available datasets demonstrate that our method achieves higher classification performance than state-of-the-art techniques. MultiChannelSleepNet is an efficient method to extract and integrate the information from multichannel PSG data, which facilitates precision sleep staging in clinical applications.