The thing with PhDs is that it often takes someone who knows the politics and people of the field in order to really understand (or help) a PhD student's troubles (if any arise). At the same time, there are many similarities.
If they don't, who are the other best people to talk to in case issues arise?
In general, university counseling should be aware of how to deal with PhD student affairs as well as undergraduate issues. (Perhaps different staff or the two groups, perhaps not.) However, I'll focus on the "if they don't" part of your question.
There is usually a graduate "officer" in most departments, who is tasked with making sure that graduate students complete the requirements of their studies, and that departmental regulations and policies are being followed. This officer should be the first person to talk to if something goes wrong, and the problem can't be resolved between the parties directly.
Beyond that, the members of the thesis committee have an obligation to intervene in the case of severe conflicts that could disrupt the program. Ultimately, though, the chair of the department would be the last "internal" stop before you would have to go to the university-level administration (the office of the dean of graduate students, or a similar position).
If you are looking for advice from fellow students, perhaps you can get information from the student committee that is present in most departments.