One atom has one proton in its nucleus. Another atom has one proton and one neutron in its nucleus. Do these two atoms belong to the same element? How do
you know?

Yes, because the number of proton are the same in both atoms

Explanation:

If an atom has only one proton in its nucleus, we can identify it as Hydrogen (H). "Another atom has one proton and one neutron in its nucleus" -- this describes Deuterium, which is Hydrogen's isotope, this means that Deuterium has the same atomic number/amount of proton in its nucleus (1) as Hydrogen, with a different amount of neutron. Recall that the no. of proton determines an atom's identity.

Deuterium is still Hydrogen because it still only has one proton in its nucleus, regardless of the no. of neutron. This makes them the same element.