Is there a positron in the nucleus of an atom?

There are no positrons in the nucleus of any atom. Positrons are anti-electrons; they are antimatter. They could be said to be the antimatter equivalent of the electron, and, as such, they would be present around the nucleus of an antimatter atom as the electrons are present around the nucleus of a "regular" atom.

Positrons can be produced in atomic nuclei by some kinds of radioactive decay, and they can be observed to be leaving a nuclear reaction called beta plus decay. But the positron leaves the nucleus of an atom as soon as it is created. It does not (cannot) exist in the nucleus of an atom.