The meaning is unclear - either of your interpretations is possible.
Bear in mind that this is likely deliberate. It's not unusual for book titles to be deliberately ambiguous or to have multiple meanings or interpretations.
The biggest source of ambiguity is the use of the word "Quest". This is both archaic and non specific - a quest can be any mission or goal one seeks to achieve, normally involving a journey somewhere. So to make the meaning clearer, we should probably replace Quest with something more precise:
Prince vs Prince: Hunt for the King
This would imply that the princes are seeking the missing King.
...Mission for the King
...Command of the King
...In Service of the King
These all maintain the ambiguity about exactly what task has to be performed, but makes it clearer that it's a task that's been assigned by the king.
...Quest for the Throne
By replacing King with Throne, it becomes clear that the Princes are competing with one another to become king themselves.