You can have a user watch particular categories (so you’d create a user with your ticket email address). Maybe you want watch-first-post. I don’t think you want a new ticket for every reply to a message?
But what you really want, I think, is a plugin that interacts with your ticketing system. You don’t mention it by name, but there are some plugins already for some ticket systems.
From the Wild Conjecture Department: If whoever is responding to tickets isn’t paying attention to the forum, then it sounds like they don’t have any reason to participate on the forum (like they get some kind of credit for closing tickets, but none for participation on the forum) and that all that matters is the ticketing system. I’d think what you want is to have whoever is attending to tickets to be getting notifications from the forum so they’ll pay attention to it.