To be safe, simply exhaust all of Iulia’s dialog options once she gets up from the bed in the Shrine of Apollo.
In order to do so, you’ll need to save Iulia by completing Dying Gasp, so do that first if you haven’t yet done so - also make sure you talk to Iulia until she gives you info on Ulpius (assuming you haven’t yet gotten the quest A Permanent Solution, you’ll know you’re ready to proceed once you get it in your log). Welcome to our quest walkthrough for A Permanent Solution, in which we convince Ulpius not to throw himself off a cliff.