Nope, not how it works.
Usually mother picks godmother, father picks godfather.
( Esme Weatherwax I swear on my tash it's daycent hash, Thu 7 May 2009, 20:09, archived)

My Godfather is called "Drinks".

( Moon Girl Technologies horrendous beanbag, Thu 7 May 2009, 20:13, archived)