My question has an inclined plane of mass $M$ and simple block kept on it, of mass $m$ (Both on a table). All surfaces are friction-less. Both of the objects would move, block down the incline and inclined plane parallel to the table, somewhat opposite to the block. Can the two equations I make be from the Free Body Diagram (FBD) of incline, in GROUND frame, and FBD of block in the INCLINED-PLANE frame? Or do I need to solve in a single frame(either GROUND or INCLINED-PLANE)?
I'd recommend that you stick to one frame, otherwise you'll have to make coordinate transformations to make your equations consistent and that's significantly more work than it's worth.
Also, be careful that if the inclined plane accelerates (which it will if I understand your setup correctly) then you have to introduce so-called fictitious forces to solve the problem correctly in the incline's (non-intertial) frame, and that's also, in this case, probably more trouble than it's worth in my opinion.