ALB re-routing from one AZ to another

Hello, our multi-zone ALB routes traffic to one node group (EKS) in one AZ. If we move one app from the current AZ to another node group in another AZ how ALB will route the traffic ? Directly to the new AZ or first to the old AZ and re-route to the new AZ ?

Depends on how your ALB is configured. An ALB doesn’t know/care which AZ your nodes are in, it routes based on configuration

We use Load Balancer controller. Which option would allow us to modify its behaviour ?

ALB only have the capability to route once, it uses the strategy you picked and goes with that one. So the question is which ALB routing strategy do you want. there are two options:
round_robin or least_outstanding_requests

In our case the target group has just one target - our app doesn’t allow load balancing across multiple pods.

If the ALB has one target, then this isn’t a question of what the ALB does, it will always route to the that target. If you want that target to something special you need to look at the configuration of how that target works, which isn’t an AWS thing.