Several machine learning-based spoiler detection models have been proposed recently to protect users from spoilers on review websites. Although dependency relations between context words are important for detecting spoilers, current attention-based spoiler detection models are insufficient for utilizing dependency relations. To address this problem, we propose a new spoiler detection model called SDGNN that is based on syntax-aware graph neural networks. In the experiments on two real-world benchmark datasets, we show that our SDGNN outperforms the existing spoiler detection models.