Is there such thing as an Availability Group failover trigger?

No. Triggers are not at an instance or AG level. You'd need to base something on the failover event or better yet, as part of a SQL Server Agent job, just check to see if that replica is the primary and then do (or not do) something.

I've done things like this before, but all I found available was the alerts. There's actually 2 different alerts, one that fires for each database in the AG, and another that fires when the entire AG fails over. So you have some flexibility.