Do you have to beat the elite four in Pokemon Emerald other make migrate option pops up Heart Gold title screen?

You don't have to beat the Elite Four in Pokémon Emerald to get the migrate option in Pokémon HeartGold's title screen.

The migrate option in Pokémon HeartGold and SoulSilver is unlocked by:

1. Connecting your DS to the internet.

2. Having a copy of Pokémon Diamond, Pearl, or Platinum saved on your DS.

Once these conditions are met, the migrate option will appear on the title screen of Pokémon HeartGold and SoulSilver, allowing you to transfer Pokémon from those games.

It's unrelated to whether you beat the Elite Four in Pokémon Emerald.