In act 1 there is supposedly a quest, but when I speak to him and choose not to pursue a romance it just ends and the quest does not complete or put itself anywhere in the journal
Small update, I've done every possible outcome with him, both romantic and non, and all I get is either approval or rivalry, and the quest never checks off
I am a bit confused. You're supposed to get friendship or rivalry for it unless you choose the very first agressive action which cut the dialog short.
Are you saying the quest still stand in your active quest under companions once you're finished?
Just leave Darktown after Tranquility and come back after visiting another part of Kirkwall. The Quest Talk to Anders will show up and can be completed without further issues.
If you talk to Anders right after Tranquility it will never appear in your Quest Log.