A: No. Totally impossible.
TL; DR: COVID-19 tests detect presence of the actual virus, which is not contained in any COVID-19 vaccine, and the vaccines cannot cause a false positive result on these tests.
COVID-19 vaccines *do not* contain any SARS-CoV-2 virus, they just teach our immune system to make antibodies against the *spike protein* of the virus. In contrast, PCR tests look for *genetic material* from the actual virus and rapid antigen tests detect presence of a *different protein* (i.e., nucleocapsid) on the surface of the virus. The vaccine *cannot*, therefore, cause you to get a positive result on either of these tests.
So, why might you test positive right after getting vaccinated or boosted?
-You might have been exposed to the SARS-CoV-2 virus right before or right after you received the vaccine and there just wasn’t enough time for your immune response against SARS-CoV-2 to ramp up and prevent you from getting infected.
-Less likely, but still possible, you could genuinely have a false positive test result, which does happen sometimes, but has nothing to do with the vaccine.
NOTE: Since COVID-19 vaccines contain no SARS-CoV-2 virus, you absolutely cannot get infected from the vaccine, so this is 100% *NOT* a reason you would test positive after getting vaccinated or boosted.
Bottom line, testing positive for COVID-19 soon after getting the COVID-19 vaccine or a booster dose *is not* and *cannot* be caused by the vaccine itself.
Stay safe, stay well.