My husband fly fishes in Oak Creek every time we go to Sedona -- at least twice a year. There are wonderful guides available through Sedona Flyfishing. I believe it's all catch and release but I'm not positive.
I don't know of any fishing in the area. If you are in Sedona for a while, you might also want to visit Prescott. Very nice, charming little town!
Yes, I believe there is a river not far from Pinon Point, just over the bridge, but you have to hike down to it. We swam in it. I am sure there are more opportunities on way to Grand Canyon area.
Get an answer from Hyatt Residence Club Sedona, Pinon Pointe staff and past guests.