There is no character named "Quaker" in Mario Kart Wii. You're likely misremembering the name, confusing it with another game, or thinking of a fan-made mod character.
The roster unlocks through Grand Prix progression and mileage:
Starters: Mario, Luigi, Peach, Daisy, Yoshi, Bowser, Wario, Donkey Kong, Toad, Koopa Troopa.
Unlockables: Dry Bones, Rosalina, Baby Mario, Baby Luigi, Baby Peach, Baby Daisy, Baby Bowser, Baby Rosalina, Mii (Male/Female).
Total: 24 playable characters + Miis.
Describe the character you're thinking of (appearance, vehicle, game) and I'll help identify it.