Killing mold IN drywall using Borax solution - what now?
NO this plan has no merit, spraying the borax water solution in a hole would only get some, if any of the mold and you would be guesting anyway. If your house was on fire would you drill holes in the walls and spray water at the holes and hope? 
The only way to check for mold is to open up the wall and testing spots that look suspicious.(If you really think that there is mold, you will need to have on a mask, googles, gloves,and full suit when opening up the wall, just to be on the safe side.) If the test show black mold you need have a specialist get rid of it. 
See number 2.
Your question leads me to believe that you think that the borax is not going to work, but to answer your question NO. If there is mold and you do not get it all out, it can come back. Mold needs to be removed and drywall replaced. Black mold need to be remove by a specialist, and then you can drywall.