No. None of the authorized and recommended COVID-19 vaccines cause you to test positive on viral tests, which are used to see if you have a current infection. Neither can any of the COVID-19 vaccines currently in clinical trials in the United States.
If your body develops an immune response to vaccination, which is the goal, you may test positive on some antibody tests. Antibody tests indicate you had a previous infection and that you may have some level of protection against the virus. Experts are currently looking at how COVID-19 vaccination may affect antibody testing results.