Seems to mostly be artificial insemination. Probably mostly due to the size differences. Although if they found the mother had to be a quail, then I suppose it is "easier" since the chicken dad just has to be small enough to not crush the lady during breeding.
I agree with you about the offspring being sterile, but the actual cross of a chicken and quail is not something impossible. It has been done "for decades".
This is from a restricted source, so I should not post the actual copy of the article, but these folks in the 1970's did it. They...