Software QA Testing is more accepting of recruits with non-engineering degrees than any other niche of the IT job market.
At Portnov Computer School out 3000 successful graduates we had many hundreds with non-engineering degrees: accountants, financial analysts, bankers, teachers, musician, business major graduates, graphic designers and other artists.
As long as you enjoy intellectual challenge, working with computers and technologies, have logical mind set - you can successfully transfer your career into Software QA Testing. It will take 3 months in the School and 2-3 months of internship to become employable. Lots of hard work and commitment are required though to be invested on your side. It is not an easy transition but it might be very rewarding.
Information Technology field is very dynamic. To grow professionally in the field over time even with formal Computer Science degree you would become the life time Computer Science student.