Yes we can, e.g. this addon does add quests about pets to pet:
https://www.esoui.com/downloads/info...andTreats.html
And Kyoma once added a custom quest for april fools, or something like that.
ppl were not ALL amused about that... Dunno why, it was fun.
Not sure though if it's still possible that way, but at least Allianym added a new tab to the quest log and added custom quests.
Perhaps adding them to the comass and/or quest trackers is harder but the journal as text would work.