What is the affiliation of the Jewish Bible Quarterly and what is its reputation in the Jewish world?
I have personally never heard of that publication; however, according to the link that you provided, it is published by the Jewish Bible Association.
The Jewish Bible Association is most famous for Chidon HaTanakh (the international Tanakh contest held every year in Israel). That contest is attended by many high school students throughout the world.
I'm not sure what exactly you mean by asking about its reputation, but it is considered a reputable organization.