Budapest private clinics (paying):
Anonim AIDS Tanácsadó Szolgálat | Karolina Rendelő
Vascular Diagnostics STI (vdiagnostics.hu)
https://belvarosiorvosicentrum.hu/nemigyogyaszat
Or google std testing Budapest. Many clinics offer such services.
Budapest public clinics:
List of places you can get tested outside of Budapest: