Do Triggers in the Active Secondary Database of an AlwaysOn Group Execute?
The secondary is readonly.  Triggers can't fire on the tables in the secondary because there are no ON SELECT triggers and DML triggers cannot be executed on a readonly database without a successful DML operation (update, insert, delete).  

Updates made on the primary and then moved to the secondary do not fire triggers on the secondary.  Instead, data changes made by the triggers on the primary are propagated to the secondary in the same manner as the direct changes made to the table.