No. The vaccination cannot result in a positive PCR or rapid antigen test result.
This is because the mRNA or DNA that enters the body through vaccination with the mRNA or vector vaccine and the protein subsequently produced are not detected by PCR or rapid antigen tests. This also applies to the protein components of vaccines. The tests detect other components of the virus.
If you are tested positive after the vaccination it means you became infected with the coronavirus before you had built immune protection – or that you have caught the virus despite the vaccination. This is because no vaccination guarantees 100% protection. In rare cases a test may also yield a false positive.