Graph representation learning (GRL) in hyperbolic space has gradually emerged as a promising approach. Meanwhile, masking and reconstruction-based (MR-based) methods lead to state-of-the-art self-supervised graph representation. However, existing MR-based methods do not fully consider deep node and structural information. Inspired by the recent active and emerging field of self-supervised learning, we propose a novel node and edge dual-masked self-supervised graph representation learning framework in hyperbolic space, named HDM-GAE. We have designed a graph dual-masked module and a hyperbolic structural self-attention encoder module to mask nodes or edges and perform node aggregation within hyperbolic space, respectively. Comprehensive experiments and ablation studies on real-world multi-category datasets, demonstrate the superiority of our method in downstream tasks such as node classification and link prediction.