How should I cite the ministry in my own country in APA?

The standard in APA when citing something titled in a language other than the one you are writing is to use the original title (transliterated to the Latin alphabet if necessary) but to also include a translation enclosed with [ ]s.

For an example from the link below:

https://blog.apastyle.org/apastyle/2018/09/how-to-quote-a-foreign-language-source-and-its-translation.html

Bussières, E.-L., St-Germain, A., Dubé, M., & Richard, M.-C. (2017). Efficacité et efficience des programmes de transition à la vie adulte: Une revue systématique [Effectiveness and efficiency of adult transition programs: A systematic review]. Canadian Psychology/Psychologie canadienne, 58, 354–365. https://doi.org/10.1037/cap0000104

See also https://guides.library.uq.edu.au/referencing/apa6/works-in-non-English-languages and https://guides.library.uq.edu.au/referencing/apa6/works-in-non-English-scripts

If the rest of your paper is in English and aimed at an international audience, it's more helpful to cite it in English translation. You might even want to write "Turkish Ministry of National Education", to make it clear which country's ministry you're referring to.

Both. If the legal/official name is X, then you certainly want X appearing, for example as a searchable item, whether or not it's in English or whatever. At the same time, yes, you probably want to give an explanatory translation (e.g., into English), for what the official title/name/label means.