'Is it possible to train a NN to detect...' - stop right there, the answer is always 'Yes.'. At least theoretically - so says the Universal Approximation Theorem. It's not a question of if you can do it, but finding out how to do it well and without burning through ten GPUs a day.
Enough being obtuse though; it's practically doable as well. You most likely want some kind of adversarial-style setup - train a recognition network to recognize a sharp image as true and the synthetically blurred one as a fake.
You could attach it to a generator network that blurs real images to more or less get an arbitrarily large training set. Really quite similar to a VAE-GAN, except you want to extract the discriminator from the GAN component.