Flamethrower is a move that is learned by various Fire-type Pokemon, but it isn't available until Generation II, which includes Pokemon Gold and Silver.
Pokemon Shiny Gold is a fan-made ROM hack of Pokemon Gold, which is a Generation II game. This means the game itself already has the ability to learn Flamethrower.
Therefore, if you're playing Pokemon Shiny Gold, you would need to find a Fire-type Pokemon that learns Flamethrower at a certain level or by TM/HM.
Here are some Pokemon that learn Flamethrower in Pokemon Gold and Silver:
* Charizard: Learns Flamethrower at level 43
* Charmeleon: Learns Flamethrower at level 43
* Vaporeon: Learns Flamethrower by TM
* Arcanine: Learns Flamethrower by TM
* Flareon: Learns Flamethrower by TM
You can find the exact details of how to get Flamethrower for each Pokemon by looking up their move sets online or in a Pokemon Gold and Silver guide.