I've been with Lily for a while and her quest hasn't triggered. She also isn't suffering from any psychotic breaks. Anyone know a way to fix this?

I will have to double check, but this really isn't a "quest" as i've read. It's more of an unmarked side thing like finding the laser pistol for the chick in the Brotherhood area. Think it happens at some random point when she has a psychotic breakdown and you need to talk to her about it. Also have you finished all of her speech functions to qualify for the unmarked "quest"?

You mean like talk about all of her topics? I might have, I'll check in a bit.

I think she has to be injured sufficiently to suffer a break.

I know, but she's gotten to less than half health and she hasn't had a break.