Initially, when asked about a quest, he will tell the Nerevarine to find out what happened to the Dwemer in Morrowind, a seemingly unsolvable task. However, this quest can be solved with the help of Edwinna Elbert.
If the player is the Master Wizard and wants be Arch-Mage, Trebonius can be killed in a duel, be forced to step down by Chancellor Ocato, or can be directly challenged if the rank of Wizard is attained.
If the Nerevarine chooses to kill him, then upon reaching the rank of Wizard, Edwinna Elbert will ask the Nerevarine to talk with Trebonius. He will then challenge them to a duel in Vivec Arena. After being killed, his clothes, staff and amulet can be looted from his body.
If the Nerevarine chooses to remove him peacefully, they can talk with Skink-in-Tree's-Shade, and she will give them a letter from Ocato promoting the Nerevarine to Arch-Mage.
Trebonius has asked you to find out what really happened to the Dwemer.
Trebonius asked you to kill all the Telvanni Councilors.
Ranis Athrys wants you to search for a Telvanni spy.
Tarer Braryn would appreciate it if you manage to convince Trebonius cure his curse.
Time for Trebonius to step down and for you to be the new Arch-Mage.