No. It will only show up if you are tested for drugs. STD testing is completely different from testing for drugs.
When doctors have you take a urine test, they analyze everything that is in your urine to detect abnormalities (Like a low number of helper T cells to indicate HIV or aids) so they can diagnose an illness. They can detect THC (Chemical compound found in in Marijuana) but, since they are testing for STDs, they won't report it or anything because of doctor patient confidentiality. So, in a nutshell, they can tell if you have if THC still in your body, but since it's not related to an illness, they wont even take note of it.
No. Totally different testing procedures.