Few-shot object detection (FSOD) aims to detect never-seen objects using few examples. This field sees recent improvement owing to the meta-learning techniques by learning how to match between the query image and few-shot class examples, such that the learned model can generalize to few-shot novel classes. However, currently, most of the meta-learning-based methods perform pairwise matching between query image regions (usually proposals) and novel classes separately, therefore failing to take into account multiple relationships among them. In this paper, we propose a novel FSOD model using heterogeneous graph convolutional networks. Through efficient message passing among all the proposal and class nodes with three different types of edges, we could obtain context-aware proposal features and query-adaptive, multiclass-enhanced prototype representations for each class, which could help promote the pairwise matching and improve final FSOD accuracy. Extensive experimental results show that our proposed model, denoted as QA-FewDet, outperforms the current state-of-the-art approaches on the PASCAL VOC and MSCOCO FSOD benchmarks under different shots and evaluation metrics.