Just one simple question. Do I need to refill the fish hatchery with more fish?

The reason I ask is because I originally put some fish in and ever few days I would get fish, but it has been about a week since the last time and still no fish. I don't have much experience with it and couldn't find the answer online.

Yes you do. I put fish in it for alchemy/potion making, and after harvesting the fish, which is either 6 or 8 of each of them, I put one of each back in and then 3/4 days later repeat. Hope that helps.