Backpacking and fly fishing sounds like a great way to celebrate your anniversary! My first recommendation would be the Rogue River. There’s a trail that follows along the river for 40+ miles, with camping areas and even lodging you can reserve if you wanted to have a night of luxury. It’s Wild and Scenic River and is absolutely beautiful — as well as a world-class fly fishing river.
My second recommendation would be the Umpqua River. This gorgeous river is also known for amazing fly fishing and beauty. It has about 70 miles of trail along the river. There could be rhododendron in bloom in early June!
The reason I recommend the Rogue over the Umpqua is that it’s a little bit farther south and may have better weather. We’ve had a huge snow year in Oregon, so I’m not sure how long it will take the trail along the Umpqua to melt out (since it’s at a higher elevation). The two rivers are probably equal in terms of fly fishing and scenery. You really can’t go wrong with either one. Both would have plenty of greenery and rugged mountains as well.