Hi Diana, to answer your question more directly, my Apple TV 3 was already logged in to my iCloud account when I enable the older two-step verification so it was already considered a "trust device" as far as my account was concerned.
When I replaced the Apple TV 3 with an Apple TV 4 I don't recall having to do anything special, I just logged in to my iCloud account without any issue. I did not have to generate an app-specific password and I did not have to append any verification codes to the password. It was a while ago so I can't remember if any other verification was required using one of my other devices.
When I changed from the older two-step verification to the newer two-factor authentication the Apple TV 4 was again already a trusted device in my account so there were no issues there either.