The grammatically correct way to spell it is bonne fête.
In Quebec, the way to wish a person happy birthday is by saying Bonne fête.
However in France, I believe that it would refer to the feast day of your patron saint.
Another View
The answer is: No, the French for "happy birthday" is "joyeux anniversaire". "Bonne fête" is what the French wish you on the the feast day of your Patron Saint. Many French people celebrate this a bit like a second birthday however, especially for children.
This said, "Bon anniversaire" and "Joyeux anniversaire" have no ambiguity.
Formal: Joyeux anniversaire Denise Informal: Bon fete Denise
In Belgian French, "happy birthday" is said as "joyeux anniversaire".
bon fete
Bon Anniversaire means Happy Birthday. Hope this helps!
Bon anniversaire a mardi!
Bon anniversaire mammy !
Bon anniversaire.
"bon anniversaire Papa"
Bon anniversaire Anthony
Bon anniversaire, beau. :)
bon sixième anniversaire
Bon anniversaire Maya!