Yes, you can. I'll give Bulbapedia's take on it since I don't know what game you're playing.
Birth Island is a location near Fortune and Quest Islands in the Sevii Islands. It is present in Pokémon FireRed, LeafGreen, and Emerald. It can only be accessed after downloading the AuroraTicket through Mystery Gift, and traveling there via the Seagallop Ferries' dock in FireRed and LeafGreen, or the S.S. Tidal in Emerald.
If you play games from Gen IV and onward, you will have to trade, hack, wait for an unlikely event or use the PokeTransfer in Gen V. It's only obtainable by event in FR/LG/E in Gen III.