It's possible, but not at all likely.
Oral sex on a female is pretty low risk for STDs, because nothing is being inserted into your throat like when someone gives oral sex to someone with a penis. Also, oral STDs are as likely to give you symptoms as genital STDs are.
When you went to the doctor, did he do a throat culture to see what, if any, bacteria was causing it? When they do that, they can also do a sensitivity test to see what antibiotic will work on it, if a bacteria is causing the infection.
Has your gf been tested for other STDs than just HIV? If she's had sexual partners before you - or if you've had sexual partners before her - it's important to test for all of them, not just HIV. There are far more common STDs, like chlamydia, that you should be tested for.
Definitely follow up with your doctor since the treatment didn't work for your throat. You may just need a different antibiotic, but if you weren't actually tested for strep, make sure to get tested.
Hi what your describing would be more in the line of what the doc diagnosed as most std from giving oral sex would be on a penis not a vagina that affects the back of the throat.