Is it a good idea to ask my Master's supervisor to help me finding a PhD in other universities?
Yes, it would be a very good idea to ask your supervisor. They know the field much better than you do. They likely have contacts in many places and also know which universities would be a good match for your interests. Many places will want a recommendation from your supervisor anyways, so it would be good to have them on board.

I would not phrase the request as "Please help me find a PhD position" but more casually, for example: "I am interested in doing a PhD, do you happen to know of any openings?".