Should I report a PhD student’s incompetence to their supervisor?

Emailing the supervisor and saying "this student is incompetent and you should get rid of him/her" would come across badly, for a few reasons:

• You lack the information to make a holistic assessment of the student.

• Even if you're right and the student is weak, weak students can improve.

• The supervisor and their institution undoubtedly have their own methods of quality control.

• Most importantly, it's not really any of your business.

It's nice that you made a good faith effort to answer the student's questions, but if the exchange has become annoying for you, your best recourse is to simply stop responding and forget the student exists.

This is unethical and unprofessional. It is simply none of your business. You are not in a position to evaluate the student. Writing to his supervisor will make you look bad. It is an insult to his supervisor.

The Ph.D. degree can not be awarded to someone incompetent. If this happens, it is going to be shown sooner or later. Let future employers and assigned assessors who are in a position to evaluate him say this, but not you. Judging a Ph.D. student requires external assessors. Let him take his time. If he is incompetent, it will be revealed sooner or later. But every student has the right to take his own time and attempts. Even the Ph.D. examination process allow several attempts before reaching such a conclusion, simply because such conclusion destroys a human's future.

If he is annoying you, you can refuse answering him. He might be wasting your time but how did you judge he is wasting other people's time? You can tell him that his questions are not reflecting the basic required knowledge and he should first build solid foundation in XYZ then come ask you. Otherwise you can not afford helping him.

Finally, we are all learning all the time. You are knowledgable in this topic he is asking but in someone's eyes ignorant in that someone's topic. You also have been ignorant about your paper's topic until you learned gradually. The world would be more peaceful if we remembered that we have not been born scientists and we learned through other noble people who gave us a hand to help us rather than a hand to destroy us, even when we have not been asked about our opinion.

NOTE: this answer was based on the assumption that OP works at the same institution and on the same campus as the supervisor in question. OP has since clarified that is not the case. I will leave the answer to stand in case it is helpful to those in a similar situation - see meta discussion.

At the risk of sounding like an old man: does no one talk to their colleagues any more? This sounds like the ideal situation for an informal chat with the supervisor. Even if you don't know them, surely there's a chance to start up a conversation after a departmental seminar or something.

Hey, your student X has been contacting me recently about their work. Sounds like an interesting project...

And play it by ear from there. It should become obvious whether the supervisor has a high opinion of their student or not. If they seem receptive, drop in that you think the student seemed to be struggling in this area. Don't charge straight in accusing student of being incompetent. If the discussion opens up then great, you can give more detail. If not, you tried, you can't force the issue.