No, sorry.
Generally speaking, feats tell you exactly what they do. If they don't say it applies to something, it probably doesn't.
Sure Climber only modifies how fast you climb on a successful Athletics check (as well as providing a slight boost to that skill).
Edit: Technically, the feat doesn't even do anything anymore. The rules for climbing were updated, and the feat wasn't adjusted to compensate. It used to be that you climbed at half-speed, now you use your full speed but it costs extra squares (like difficult terrain, but not counting as difficult terrain).
If one of my players chose to take the feat, as DM I'd rule that it removes the extra cost (making it functionally almost identical to how it used to work). That's something you'll have to discuss with your DM (or rule on, if you are the DM).
