PhD Research under guide/advisor of a different department
Yes.

At least in the US, it is quite common for PhD students enrolled in department X to be advised by faculty in department Y, either alone or with a co-advisor in department X.  Off the top of my head, I can think of PhD students in my (computer science) department who are advised or co-advised by faculty in electrical engineering, mathematics, and industrial engineering.  Similarly, faculty in my department who advise PhD students in mathematics, electrical engineering, mechanical engineering, and biology.

That said, if you enroll in the PhD program in department X, you will have to satisfy the degree requirements (courses, comprehensive/qualifying exams, etc.) for department X, even if your advisor and your eventual research interests are in department Y.