I have not fished Western NY, but have driven through it more times than I can count over the years. Elicotville is a beautiful place, especially in the winter. It's been a long while since I was last there, but Holiday Valley always got very good snow.
I am more familiar with PA waters being that I'm a resident. At least on this side of the state line, you will find a variety of fish in the Allegheny itself, trout included. The tributaries in PA are also known to hold trout either wild or dumped in by sportsmans clubs or by the PA Fish & Boat Commission's "Great White Fleet" of trucks.
The NY DNR page would be a very good place to start your search, these pages are often overlooked.
Fishing - NYS Dept. of Environmental Conservation
I do agree with previous comments to consult with a fly shop and perhaps even hire a guide for a day to get you familiar with the lay of the land so to speak.
Good luck and enjoy that chalet!