Accurate image feature point detection and matching are essential to computer vision tasks such as panoramic image stitching and 3D reconstruction. However, ordinary feature point approaches cannot be directly applied to fisheye images due to their large distortion, which makes the ordinary camera model unable to adapt. To address such a problem, this paper proposes a self-supervised learning method for feature point detection and matching on fisheye images. This method utilizes a Siamese network to automatically learn the correspondence of feature points across transformed image pairs to avoid high annotation costs. Due to the scarcity of the fisheye image dataset, a two-stage viewpoint transform pipeline is also adopted for image augmentation to increase the data variety. Furthermore, this method adopts both deformable convolution and contrastive learning loss to improve the feature extraction and description of distorted image regions. Compared with traditional feature point detectors and matchers, this method has been demonstrated with superior performance on fisheye images.