I'm pretty sure that example only uses Gaussian noise, which may not be easy to change to Poisson. You may have to set up sample images yourself. Try following this example to set up the training image structure instead.
I'm pretty sure the key is to set up a randomPatchExtractionDatastore that will contain both your input and output images to the network.
Then you'll want to set up the network. I haven't looked into how the network is set up for denoising, so you'll want to see what the example is doing and perhaps copy that.
Let me know if I can help further.