We've seen a few reports from PlayStation users who are unable to accept the initial quest in Coerthas even though they've registered the code and patched the game. However, it seems the cause is due to the fact that the Heavensward software was not downloaded from PlayStation Network. This is a step that cannot be skipped and is likely what is causing the issue.
If you're finding yourself unable to accept the first quest in Heavensward though you've registered your code and see early access on Mog Station, please ensure you've downloaded the Heavensward installer file from PSN.
PlayStation®4 users
Navigate to your Library and find FINAL FANTASY XIV, select it and go to "Related Items" - the Heavensward software will be there for you to download.
PlayStation®3 users
Navigate to PlayStation Network on the XMB, select Account Management and then Transaction Management. Select Download List and you should see the Heavensward software listed and you can download it if you not yet done so.