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.
bicycles.stackexchange.com/users/6782/BPugh
I have a Chariot (now Thule) cart attached to my Brompton (with internal gears).
It has the standard adapter (similar to the one in the picture above) and works fine.
I didn't have to do any adaptations.
The only issue is that the cart leans forward due to the 20" wheel but my little one does just fine.
bicycles.stackexchange.com/users/24183/JoseCarlos
How much "like" does it have to be?
I have seen trailers that clamp onto the seat post.
These have the added advantage that when unhooked from the bike, you can use it like a hand truck.
I used to go grocery shopping with a Burley axle-mount, and had to rearrange the groceries when I lowered the tow-bar to axle height.
In one model I saw, the cargo compartment was easily removable from the frame and had straps to make it a backpack.
UPDATE: I now have a Cyclone IV Chubby .
On the road, my supplies go inside it.
When I get to an airport, I empty it, fold the Brompton and put it inside.
Surround it by whatever else will fit and check that in.
The rest goes in the backpack for carry-on.
Downside:
excess baggage fees Radical Design makes other trailers.
This person uses a child trailer with Brompton
bicycles.stackexchange.com/users/13605/WGroleau
This should work.
I use the Thule Chinook 1 with my Brompton folding bike.
The only inconvenience is that the front wheel is best not folded down but removed.
The clearance with the ground is too low.
But when you remove the front wheel, it works perfectly.
Based on this experience I would say yes!
bicycles.stackexchange.com/users/26197/Christopher
Yes i can confirm that a Thule Chariot goes great on a Brompton.
The clearance from the road to the coupling is clearly lower than on a larger-wheeled bike but it handles well and has not slowed us down at all.
bicycles.stackexchange.com/users/46920/Zadi
I wonder if I can use a Brompton bike with a trailer like the Thule Chariot Cougar? It uses the so called  Thule Axle Mount  and I wonder if it is compatible to the bike. At this point, it is just advanced planning and I can't test the possible setup. What do you think, any experiences?
Can I use a Brompton bike with a trailer like the Thule Chariot Cougar?
bicycles.stackexchange.com
<trailer><brompton>
bicycles.stackexchange.com/questions/31288
bicycles.stackexchange.com/users/11069/lejonet
Many cyclists confirm that a Thule Chariot goes great on a Brompton. There may be a few little issues to deal with, like slight leaning forwards, but nothing major. Yes, this is possible, but the weight should be adjusted to prevent the cart leaning forward too much. The front wheel should also be removed rather than folded away due to the low clearance. Other trailers that clamp to the seat post may be easier.
Yes, this is possible, but the weight should be adjusted to prevent the cart leaning forward too much. The front wheel should also be removed rather than folded away due to the low clearance. Other trailers that clamp to the seat post may be easier.