If you wish to prove the authenticity of a document issued in Hungary in order to use it in Turkey, Apostille authentication is required.
For Hungarian documents, the Apostille certificate is issued by the competent Hungarian authority (Ministry of Foreign Affairs and Trade, Ministry of Justice, Hungarian National Chamber of Notaries) in Hungary.
Consulates or embassies are not competent in issuing Apostille certificates (nor is the Consulate General of Hungary in Istanbul). You can read more about the Apostille certificate here: http://konzuliszolgalat.kormany.hu/felulhitelesites
In order to use a Turkish document in Hungary, the competent Turkish authority issues an Apostille (the Consulate General of Hungary in Istanbul is not competent).