.dwg format has never heard of Zones. So want do you want to achieve? If you export the fill to dwg it will not do any good since Autocad measures area by boundary or polyline and it has no connection with the text of the on the screen.
I'm doing a feasibility study for an office building . I don't want to measure the areas in AutoCAD (since I already measured them in ArchiCAD). I just want the appearance of the exported drawing to resemble the appearance in ArchiCAD (so that I can send it out to the client or consultants). I have different zone colours for the offices that belong to different departments of the company that is going to use the office building. The colours all completely messed up, most of them turned to the same gray colour so I can not distinguish different deparments any more.