Is it acceptable to look up a student on the Internet to find out whether I have a conflict of interest?
First of all, let’s get something out of the way. You, and me, and everyone reading this, has on occasion googled the name of someone we know - a friend, colleague, student, professor or whatever - to see if we can find out more about them. Some of us do it more frequently than others, and some of us are fine doing it out of sheer curiosity while others feel a need to rationalize their behavior with arguments about trying to avoid conflicts of interest... but all of us do it. So, if it’s “stalking”, then everyone’s a stalker, and the word “stalking” ceases to have any meaning. Which would be a shame, since there are in fact genuine stalkers out there, and it’s sadly necessary to have a word to describe their behavior.

Now, getting to your specific question: you ask about ethics, but this simply isn’t an ethics issue. You can ethically google whomever you want whenever you want. It’s what you do with the information you obtain once you have it that may or may not be ethical. It may also be inadvisable to google your students, or generally to develop the habit of googling lots of people, for reasons unrelated to ethics. Personally I try to keep my people-googling to a minimum so that I don’t walk around feeling that I’m a nosy person who gets into other people’s business that doesn’t concern me. (I also worry that I might find something I would be better off not knowing, and then I’ll have to either pretend that I don’t know it, or admit to the person that I know it because I googled them - pretty awkward either way...). In any case, I don’t think it’s unethical, and if you feel like you have some good reason to do it, I think it’s perfectly fine.