I talk to the npc and I get the message that appears in the image, it does not allow me to travel to the tower. I want to get to be an adult and get the subclass.
In the image you can see that I have the mission completed to talk to the npc and travel to the tower.
After you talk with NPC after you started Adult Quest you need to do quite short quest line and that quest line starts near that NPC at other NPC. Just check the video I dropped below: