To address the morphological processing problem of images on quantum computers, several morphological processing methods for quantum images were studied. Firstly, an improved quantum representation method for images with any size was proposed. In the method, both pixel value and pixel position were represented by quantum basis state, and the number of quantum basis states representing pixel positions was equal to the number of pixels. Then, by designing quantum circuits for the two basic operations of dilation and erosion, several quantum morphological processing methods for binary images were designed, including noise removal, boundary extraction, and skeleton extraction. Finally, the implementation effect of the designed methods was verified through simulation on classical computer, and the complexity of quantum circuits was analyzed based on the number of basic quantum gates used. The results show that the methods proposed in this work can achieve speedup over classical methods.