I remember reading through lists of Q names in baby name books and online sites, always thinking "there has to be more." Just for example, there are 14 Q girls names and 36 Q boys names on Behind the Name, and 22 Q names for girls on Nameberry. Not too many for girls, and in comparison with all the A names out there, well...
Thanks to White Pages and 100,000 Baby Names by Bruce Lansky, I've found so many more. There are literally hundreds, although not all of them may be the original first form of a name. Take your pick, then research it elsewhere. Here's a small sampling of girls (I'll go with triple the amount shown on Behind the Name).
Quilla (Mama Quilla was an Incan goddess)
Quenilda (featured on Nook of Names)
Quarralia (Australian, star)