I have seen some of that information on the USGS websites but it seems you need to go look for it. It's under historical data or something like that.
I also have a book titled "Western Whitewater" by Jim Cassady, Bill Cross and Fryar Calhoun that has average seasonal flow profiles for many rivers throughout the western United States. Copyright is 1994 so the book may be dated but that average flows and when they peak will be pretty good.
Hope that helps