The periodic table tells us that there are 6 protons in a carbon atom. Is there a way to verify this first-hand? Or are we just expected to believe it unquestioned?
1) While not easy, it is possible to obtain mass-spectrum of full range of ions, including fully ionized. By number of peaks with Z/E corresponding as X, X/2, X/3 ... X/n it is possible to ensure that an element has exactly n electrons and protons. It is still difficult to ensure full ionization of heavier atoms, so the method is not applicable for heavier elements. This, however, is the only direct method I can think of. Should work for carbon, though.
2) Various X-ray-derived spectral methods. While not providing direct evidence, they do provide information on electronic shell structure, that can be compared with theoretical calculations.
Carbon is defined as an atom with six protons (and 6 electrons). That is the nature of the periodic table. It is so per definition. I suppose your real question is: how can we determine if something is really carbon. Right?