Answer
No, hydrogen cannot emit x-rays.
Work Step by Step
The energy of the ground state of a hydrogen atom is -13.6 eV. Even if an electron dropped from $n=\infty$ down to the ground state, the emitted photon would only have energy of 13.6 eV, which is not energetic enough to be an x-ray.
X-ray targets are generally composed of higher atomic number metals.
