You can do the driving licence training at any driving school that is regulated and recognized by the government.
Here you can find the webpage of the Nemzeti Közlekedési Hatóság (National Transportation Authority) which is the responsible institution for the examination:
- (in Hungarian): http://www.nkh.gov.hu/a-hatosagrol
- (in English): http://www.nkh.gov.hu/en/web/english/
Here you can find more information about getting a driving license in Hungary (in English): http://www.budapest-moms.com/2011/01/how-to-get-your-driver%E2%80%99s-license-in-hungary-a-guide-for-non-europeans/