I'm asking because I thought they did, but just looked at the terms and conditions:
https://www.clubcarlson.com/section/...ions.sidemenus
The benefits page says most things apply on Eligible Stays, and that refers to Eligible Rate, which is defined as "a rate identified in Carlson’s reservation system as eligible for Gold Points".
I assume that "eligible for Gold Points" means you can earn points on that rate. Which would exclude award stays.