Few-shot image generation aims to generate diverse and high-quality images for an unseen class given only a few examples in that class. However, existing methods often suffer from a trade-off between image quality and diversity while offering limited control over the attributes of newly generated images. In this work, we propose Hyperbolic Diffusion Autoencoders (HypDAE), a novel approach that operates in hyperbolic space to capture hierarchical relationships among images from seen categories. By leveraging pre-trained foundation models, HypDAE generates diverse new images for unseen categories with exceptional quality by varying semantic codes. Most importantly, the hyperbolic representation introduces an additional degree of control over semantic diversity through the adjustment of radii within the hyperbolic disk. Extensive experiments and visualizations demonstrate that HypDAE significantly outperforms prior methods by achieving a superior balance between quality and diversity with limited data and offers a highly controllable and interpretable generation process.