The element with 12 protons is Magnesium (Mg).
Its energy shell electronic configuration is (2,8,2)

Its valency is 2. So, it tends to lose 2 electrons instead of gaining more electrons. So, it is virtually not possible in our lives, under normal circumstances.

So, there cannot be 12 protons and 13 electrons in a single atom.

However, everything is possible in Nuclear Chemistry. Unfortunately, I'm not too well versed with it. But you may get a "Yes" answer only from a Nuclear Chemist.
The element with 12 proton and 12 electron is magnesium (Mg).

Magnesium metal losses two electrons to get octet in its valence shell.

Magnesium do not gains a electron to have 13 electrons which is unstable.

Therefore element with 13 electron and 12  proton is not possible.