That's a great idea! Are there any how to photos?

Where ever you can find these sorts of rocks in your area should be able to tell you how to put together and even have all the supplies - filter, tubing, receptacle to catch and recirculate the water, etc.

This link explains them pretty well :https://www.aquascapeinc.com/fountain-kit

Similar to gardengal's picture, a waterfall with a rock bed instead of a pond. Birds love it.

Pondless Waterfalls, Water Features Rochester NY · More Info

See that picture has a lovely corner to build that in, I have an ugly, flat flower bed with an open yard on one side and a patio behind it. I think Gardengal has hit the nail on the head. With a water feature like that, it's good for birds and can be enjoyed by everyone no matter where they are sitting.

Just fyi you will need power to run the pump for the water feature. I didn’t see an outlet in the flower bed?

If you put a fountain with rocks out in the sun, you will be forever battling the algae that grows on the rocks. I had a similar fountain, used all the algae killer recommended, and still had to take the whole thing apart every month, scrub it down and re-assemble. I recommend something without rocks that is easier to keep clean. I'd buy a concrete or metal bowl fountain if I had to do it over again. Not resin, it doesn't hold up.

Never have had any issues with algae growth on these sorts of self contained water features. Some develops on the major feature, yes, but so it does on rocks in a river as well. Just adds to the natural appeal, IMO. Never an issue on the surrounding river rock, as the water doesn't remain present for any length of time and they dry off rapidly.

Wouldn’t any algae and moss be a bonus? Of course I live in the land where everything is covered in moss. Lol

Well, myself and my friend both had this problem with our home pond and fountain in sun. I also maintain three ponds at work and scrub the rocks on them constantly plus treat with an algicide. YMMV.