Quantum machine learning is a promising direction for building more efficient and expressive models, particularly in domains where understanding complex, structured data is critical. We present the Quantum Graph Transformer (QGT), a hybrid graph-based architecture that integrates a quantum self-attention mechanism into the message-passing framework for structured language modeling. The attention mechanism is implemented using parameterized quantum circuits (PQCs), which enable the model to capture rich contextual relationships while significantly reducing the number of trainable parameters compared to classical attention mechanisms. We evaluate QGT on five sentiment classification benchmarks. Experimental results demonstrate that QGT effectively learns sentiment representations, achieving accuracies of 93.0%, 90.5%, and 88.0% on the Yelp, IMDB, and Amazon datasets, and 100% and 95.12% on the synthetic MC and RP datasets, respectively. These results highlight the potential of graph-based quantum NLP techniques for advancing efficient and scalable language understanding.