I'm not sure it's about rubbish, as such --- I've heard these things also referred to as 'vide poches', and they're just those useful compartments for "putting things in" (loose change, old sweets, phones, glasses, etc.)

I don't actually know if an exact term even exists in English, but I would suggest 'console', as they usually form some sort of 'box' between the front seats, and sometimes contain window, heating or other controls.

If you think 'console' sounds too pretentious, you might try 'stowage compartment'.