Upgrade Your Education
Look for continuing education courses which will allow you to gain knowledge in those fields.
A lot of colleges and universities offer night or online classes regarding computer programming / networking / testing basics. Getting a few of those under your belt might be enough to land you a junior position in the field you're interested in.
Ask For Training
Does your current company have a QA department? Ask your manager for the opportunity to do a little cross training.
If you do not have a a QA department then ask your manager to allow you to perform some QA testing on some company projects before they are rolled out. He might appreciate your initiative, and you could use that bullet point on your resume.
Consider getting involved with an open source project with requires some low-level QA work and getting experience in this way. Then you can pad your resume with relevant experience. This will work best when done in tandem with the above.