For most of the UK there's a website that offers a postcode search facility for people to find dentists in their area, with an indication as to which of those practices (if any) are currently taking on new NHS patients.
However Northern Ireland is, as usual, different. All that's available online is a complete list of all dental surgeries in NI (in the form of a downloadable Excel spreadsheet) and the rather unhelpful statement that "If you are looking for a Health Service dentist you can look at Health and Social Care Board Dental List at their headquarters or local offices". (There are email addresses and telephone numbers given for the relevant offices though, so you could try contacting them that way).
https://hscbusiness.hscni.net/2070.htm