I don't like a loud clicker on my reels either.
I have several of the Allen trout reels, the old version, and they do not click.
http://www.allenflyfishing.com/trout-ii-reel-series/

I have lately been using a Ross Flyrise reel, and I really like it better than any others I have tried. It has a faint click, I don't mind it at all. It is a little cheaper then the Allen, but Allen does run sales from time to time.