Can atomic number change the type of electromagnetic radiation being emitted when a electron drop from a energy level to another?
Absolutely, the differences in potential energies will be different for every element. This is because the potential energies will be different for every element. Notice that the potential energy for each energy level can be determined from the ground state. Let $E_g$ be the potential energy of the ground state (with the sign). Then for energy level $n$ the potential energy is $\frac{E_g}{n^2}$. Thus Differences between energy levels $n$ and $m$ can be written as $E_g\left|\frac{1}{n^2}-\frac{1}{m^2}\right|$.

Clearly it will be different for a different ground state energy which is precisely what the number of protons affects.

Note that the only important part of the mathematics is that all differences rely on the ground state potentials.