How to level depressed region in plywood subfloor for laminate?
First you need to address and fix the sagging issues before proceeding with any thoughts of next steps. Get under the house and see what's going on. If it is not severe (ie. rot/mold) then get proper bracing in there. 

After that, you should most certainly be dealing with a much more level floor. As for liquid level vs. replacing the subfloor. It is much more cost effective and ideal in laying fresh particle board on top and have a nice new surface to start with.  Liquid level will work well when no more than 1/4".


  Could I use 1/4" thick plywood pieces for this? If so, are there additional steps besides nailing it down?


I would go with at least 3/8" particle board and as for nails, DO NOT USE any. Screw it all down, or the nails can get loose over time and make the floors squeak. 

With that, make sure to screw off all areas that squeak, or you will live to regret it.

Update Based on it is now a Second Floor Condo.
Are both floors your condo and do you know much about history? 

I apologize if this type of reply is not allowed. As I will remove this portion or answer in full depending on info to come.