You could, but why would you want to do that? The process of creating Bremsstrahlung is spectacularly crappy when it comes to energy efficienc as 99-99.8% of the energy you put it just gets converted to heat. And you don't even get a good signal as the energy of the emitted xrays isn't one exact energy (okay, for the most part) but is a wild bunch of different wavelengths, ranging from 0.2 to over 1 Angstroem (for a Wolfram anode@50kV).
How and why that's the case: Bremsstrahlung is the result of an electron missing an atom (the nuclei) and releasing its energy in the form of photons as it changes its path (it gets attracted to the nuclei as the nuclei has a positive charge and the electron a negative). That's why you get a energy spread that accords to Kramer's law as the energy of the Bremsstrahlung is related to the distance between the nuclei and the electron and its speed->there is an infinite amount of possible distance/speed combinations.
An electron can also just happen to bump into an electron that is just sitting inside an atom orbital of an atom: If that happens, the electron that was bound to the atom gets released and both electrons fly away (the energy of the first electron is reduced by the amount of energy it took to free the second electron). If the atom has another electron that has a higher energy as the one that was just released this third e- will take the place of the second, releasing the difference in their energy as one photon.
For example: The first e- has the energy 1 and it takes 1 energy to release the second e-. They collide-> second e- is released. The third electron has an energy of 2 as it has to occupy a atom orbital with a higher energy (the lower energy orbital was occupied by e- two). That electron jumps from its orbital to the one where e- two just was, releasing 1 energy in the process.
Now this energy only depends on the atom and the orbitals of the two electrons involved: If we have a copper atom and the electrons goes from the orbital with the second lowest energy (L) to the orbital with the lowest energy (K), we call it the K-alpha radiation of copper. K because thats the orbital were electron three goes, alpha because the electron came from the orbital with the next higher energy (beta for the orb after that etc).
If we want to use xrays as a form of communication we should go with that approach as we can just pick an element that doesn't occur naturally as our cathode and every signal of that specific wavelength can only come from our space probe. There is just one tiny problem: generating Bremsstrahlung is extremely inefficient (and the amount of the K alpha radiation we want is even lower), you'd need a power supply in the kW range (solar can do that, but not for deep space probes). The second, more important problem is that there is no way to cool the cathode as the only way to cool an object in space is to just let it radiate the heat as black body radiation and that is painfully inefficient compared to air/water cooling on earth. Just look at the ISS, those massive white panels are there to keep it cool and they need roughly 6m^2 of panels for every kW of heat dissipation.