Probably not since they are different species but then again you can even cross guinea fowl and chickens. It's just very unlikely to happen. If you did get offspring they'd probably be sterile hybrids. You'd also have to do artificial insemination because they'd be unable to mate if they even thought to attempt it.
i think they could personally, they're both from the "coturnix" family after all....and I say if they can hybrid quail to chicken, pheasant to chicken, chicken to peacock or whatever they've all already done,then why not BQ to Coturnix. I want to try it to incorporate the friendliness of the Coturnix, and the various colors of the BQ into one hybrid.