Ambiguity is inevitable in medical images, which often results in different image interpretations (e.g. object boundaries or segmentation maps) from different human experts. Thus, a model that learns the ambiguity and outputs a probability distribution of the target, would be valuable for medical applications to assess the uncertainty of diagnosis. In this paper, we propose a powerful generative model to learn a representation of ambiguity and to generate probabilistic outputs. Our model, named Coordinate Quantization Variational Autoencoder (CQ-VAE) employs a discrete latent space with an internal discrete probability distribution by quantizing the coordinates of a continuous latent space. As a result, the output distribution from CQ-VAE is discrete. During training, Gumbel-Softmax sampling is used to enable backpropagation through the discrete latent space. A matching algorithm is used to establish the correspondence between model-generated samples and ground-truth samples, which makes a trade-off between the ability to generate new samples and the ability to represent training samples. Besides these probabilistic components to generate possible outputs, our model has a deterministic path to output the best estimation. We demonstrated our method on a lumbar disk image dataset, and the results show that our CQ-VAE can learn lumbar disk shape variation and uncertainty.