The Red Cross phone number for Belfast appears to be 028 9073 5350. However that's for their training centre, so they might not be able to assist with transport enquiries. (It could still be worth a try though, as they might be able to suggest who to contact).
The Disability Action Transport Scheme operates in and around large towns in Northern Ireland, helping people who're unable to use public transport to get to medical appointments (as well as for social, shopping or work-related travel). They charge just £2.50 each way:
https://www.disabilityaction.org/transport