"Fête" is actually used to refer to the French (catholic) observance of a person's "saint's day", also called "name day". The idea is that each day in the year is associated with a different saint. For those who observe this practice, it is a special celebration for the day associated with the same name as yourself. https://en.wikipedia.org/wiki/Calendar_of_saints It might be mistaken for a birthday, but it is actually a saint's day.
Indeed, "Bonne fête à toi" is something French Canadians might sing at a birthday, and the following Wiktionary entry includes "(Canada) birthday" as one of the definitions of "fête":
"Fête" for this usage is probably a result of the confounding of the name-day celebrations mentioned by n6zs with birthday celebrations, particularly since French-speaking Canada is traditionally very much Catholic.
But "anniversaire" would still be valid for "birthday" in French-speaking Canada, and strictly speaking might be considered more correct. I have a Canadian dictionary that lists "birthday" for only the "anniversaire" entry in the French-English portion, not for the "fête" entry, but in the English-French portion it lists "[saint's day] fête" for the "birthday" entry.
But keep in mind that "fête" can also be just a celebration in general.
It's okay according to Collins:
What you're asking about is called "elision" or "contraction", not "conjugation".
In French, elision doesn't take away consonants (like the two ls in "quelle"), so you would never see it work as you suggest.
You can read more about it at these sites: