To all Doors/Quarters need do the "In Service of Yalahar Quest" while you advance on missions you gonna unlock the doors.
You don't need completed all the quest to gain all access. Every mission you start unlock a new door but it need be do in order. By example you cannot unlock the "Sunken Quarter " before do all first 6 missions.
Access to Alchemist Quarter mechanism - Start mission 3.
Access to Trade Quarter mechanism - Start mission 4.
Access to Arena Quarter mechanism - Start mission 5.
Access to Cemetery Quarter mechanism - Start mission 6.
Access to Sunken Quarter mechanism - Start mission 7.
Access to Factory Quarter mechanism - Start mission 8.
https://tibia.fandom.com/wiki/In_Service_of_Yalahar_Quest/Spoiler