In general, documentation requirements & underwriting rules for Euro "green card" insurance that apply to motorcycles apply to all motorvehicles (incl. cars). Rates may vary of course.

"Green card" usually refers to minimum required third-party liability only, not "comprehensive," (damage to vehicle). Neither the Luxembourg agent for ADAC referenced above in this thread, nor any of the Euro auto clubs (German ADAC, Italian TCI) will underwrite "comprehensive" insurance as part of the "green card" policy they issue for non-member/ citizen tourists.

For non-citizen/ members, Euro "Green card" coverage is usually issued thru national auto clubs or their direct agents. I have never heard of green card cover being issued by the Belgian auto club/ Bruxelles, but you could ask directly Royal Automobile Club of Belgium

For your US car, if you need comp, in addition to Euro green card liability, you need to get it thru a specialty broker in the US. There's a list of specialty US brokers here:
BMW MOA : Global Touring - to Europe,
(page down to " EURO INSURANCE")
Most of the specialty Euro/overseas insurance for US-registered vehicles is underwritten by AIG (the "solvent" part of the company )
You'll want to have at least a minimum Euro green card insurance document in hand when you pick up your car from the Belgian port dock, because Belgian Customs may not release the car without it; and in any case you don't want to be driving it w/o min cover.