This looks like a great article on how to start to transition from SUP on flat water to SUP surfing. For some it might be an easy transition but for most it proves to be challenging.
Beginner Series | Learning How to SUP Surf
By Jack Haworth
Contrary to popular belief, SUP surfing is not easy.
My first foray into SUP surfing ended with humiliation. I could barely manage to balance on my board–much less actually catch waves–and eventually got tired of falling and paddling back to the beach in shame. After spending more than a decade longboarding, I had mistakenly assumed my transition to SUP would be a breeze. I was wrong.
That day I learned SUP surfing and prone surfing are two different beasts and the sooner you accept that, the quicker you will improve. So take a deep breath, reign in your pride, and prepare to embarrass yourself during the first couple attempts.