I fail to see why it wouldn't be possible. I assume a PhD in climatology would require 1. an understanding of the field and its state and 2. being able to perform research in the field.

I'd certainly expect somebody serious about climate change (including disagreeing with its existence or with it being anthropogenic) to be able to understand what it's about in the first place, how climate model work, how to create new model matching available data or mine for new data, etc...

And thus I see no reason why they wouldn't be able to get a climatology PhD.