In my idiolect, both "Я не понимаю этот вопрос" and "Я не понимаю этого вопроса" are correct. But "Я не понимаю этого вопроса" sounds bookish and "Я не понимаю этот вопрос" sounds as in everyday conversations.
I do not claim that my opinion is absolutely true. If you've spotted any mistake in my English, please, correct it. I want to be aware of any mistakes to efficiently eliminate them before they become a habit.