If you migrated from PS3 to PS4 then you have the digital download version of FFXIV (no disc) so as long as you sign in using the same PSN username after putting in the new hard drive you'll be able to download the game again as it's already in your account.

You will have to redo all your gearsets and settings though sorry. That stuff is stored locally. I've personally never looked at how the PS4 formats it's hard drive, but I doubt it's going to be something simple to move them across