I didn't even quit the employer in question, a temp agency. After an assignment at a company in which my hours were being altered illegally, I chose not to call the temp agency and seek work based on their lack of action in correcting abuses. Instead I obtained a better paying job no more than a week and a half later. I worked for this new employer for a month and a half and was laid off. I then filed for unemployment. I was laid off for a little less than a month and am working again with this employer. I did not claim unemployment for the week or so after the temp agency. I claimed after being laid off from a new employer.
Employment / Labor Attorney
I have handled many appeals from denials of unemployment compensation. I have never seen a single instance of denial based on why you left "prior" employment, it focuses on the reason employment with your last employer ended. Therefore the answer is "no," if by "based on a 'prior' job," you mean will they question whether your quit that job, or not, or why.
The job prior to the last one would enter in only in determining if you have enough work credits to qualify for benefits.
If you are denied benefits, contact an attorney about an appeal.