Provided you haven't already completed it, you need to progress the main story by completing Venesh/Sarrina's quest by speaking to the butler in your home base, either rescuing Rose with Venesh or freeing the person that is imprisoned in the Crusader base with Sarrina. Once you've done that and finished exploring around the areas you unlock from doing so, going back to your home base will cause a message from the butler to pop up to speak to Claude which moves the storyline forward.
If you've reached the point after beating the demon that there's the king's messenger waiting back at your home base in the main room and the warning of "proceeding further will lock out other options", then you've finished Claude's quest.