Graph neural networks (GNNs), grounded in spatial or spectral domains, have achieved remarkable success in learning graph representations in Euclidean space. Recent advances in spatial GNNs reveal that embedding graph nodes with hierarchical structures into hyperbolic space is more effective, reducing distortion compared to Euclidean embeddings. However, extending spectral GNNs to hyperbolic space remains several challenges, particularly in defining spectral graph convolution and enabling message passing within the hyperbolic geometry. To address these challenges, we propose the hyperbolic graph wavelet neural network (HGWNN), a novel approach for modeling spectral GNNs in hyperbolic space. Specifically, we first define feature transformation and spectral graph wavelet convolution on the hyperboloid manifold using exponential and logarithmic mappings, without increasing model parameter complexity. Moreover, we enable non-linear activation on the Poincaré manifold and efficient message passing via diffeomorphic transformations between the hyperboloid and Poincaré models. Experiments on four benchmark datasets demonstrate the effectiveness of our proposed HGWNN over baseline systems.