An Airbus is not a Boeing. If I understand what you're asking, no you cannot show your A320 rating and get a 777 rating. The only thing similar between the two is that they're both Jets.
For someone like myself, to go from the 737NG to the 777 is I think 3-4 sim sessions.
Now, you can of course train to become a 777 Pilot, ideally not at your own expense
-but- many Airlines now days hire Type rated or "similar" experience Pilots, which usually means Airbus -> Airbus or Boeing -> Boeing.