I'm tying to find out how Morte died but so far I have only had him reveal his travels with my past incarnations. Is there a quest, area, or event I need to find, complete, or trigger in order to gain his confidence? My location is at the end of the game, before I enter the Fortress of Regrets. As such I have already discovered Dak'kon's past if that helps.

In order to get his stat boost you have to talk with Fall-from-Grace about him and confront him with what you've heard.

Also, did you confront Morte about the "last line"? If not go back to TNO Crypt and read the tablets.

As I recall, he have little knowledge of his life as a meatbag, but after visiting the Pillar of Skulls Nameless One should have additional option in dialog with Morte. There you could recall a memory of the Practical Incarnation being very, very angry about Morte not knowing something he claimed knowing.