Water LabyrinthFireRed&LeafGreen
"A small patch of open water between Five and Six Island. An NPC here gives you a Mystery Egg that hatches into Togepi, one of the only ways to get one in FRLG."
WATER LABYRINTH MAP
Open water area west of Five Island. Surf through the channels between small islands. No real maze despite the name, just winding waterways.
Pokemon Encounters
A free Togepi at level 5 from the NPC on the island, the only way to get one in FRLG. Hoppip at 30% while surfing and standard Sevii fishing round out the catches.
Togepi
TRAINERS
A Pokemon Breeder near level 48 on this short water stretch. Mixed team, over quickly.

Alize
Water Labyrinth Guide
A man on Water Labyrinth (the small island between Five Island and Resort Gorgeous) gives you a Togepi Egg if your lead Pokemon has maximum friendship. Water Labyrinth is the only source of Togepi in FRLG. The egg hatches after walking, and Togepi evolves into Togetic via high friendship.
Islands One through Three unlock after you beat Blaine on Cinnabar Island. Bill takes you there as part of the story. Islands Four through Seven require beating the Elite Four first, then completing Celio's Ruby and Sapphire quest on One Island to get the Rainbow Pass.
You can physically reach all seven islands without it, but evolved Johto Pokemon only show up in the wild after you get the National Dex. That means 60 or more Pokedex entries plus completing Celio's quest. If you're only seeing Kanto species, that's why.