Can I use a Brompton bike with a trailer like the Thule Chariot Cougar?
I'm going to say yes based on this photo.  It looks that the hole is big enough to pass over just about any axle, quick release or nutted.  I think what is throwing you off is that it comes with a quick release skewer that is longer than what is typically used for wheels.  You may not need that provided skewer.  Burley uses a similar one and I was able to put the hitch on a Jamis bike using the skewers that came with the bike.  

If your bike has the internal gear hub, it probably has anti rotation washer (special washers to keep the axle from turning).  This hitch would mount to the outside of that washer.