Is "What's your Islam question?" grammatically correct?

Yes, nouns function attributively all the time in English:

• This chicken soup is delicious.
• I have a marriage problem.
• He’s a football player.

Or even

• Try this new dessert — it’s like a doughnut croissant.

This makes compound nouns out of the pair. Neither is an adjective.

This use of attributive nouns actually distinguishes English from many Romance languages. For example, in Spanish, you could not say What’s your Islam question? Rather, you would have to say What's your question about Islam?

In modern Romance, because case markers no longer exist on nouns, we normally use some preposition to separate the two nouns instead:

• A football player is un jugador de fútbol.
• An Islam question is una pregunta sobre Islam.

Although it’s far less common than in English, even in Spanish nouns can occasionally function attributively to modify another noun:

• A police dog (those are both nouns) is un perro policía (and so are those).

In Latin, the word for the noun Islam would be marked in what’s called the genitive case to mark it for attribute use, as in Islamus > Islami. You could also sometimes use actual adjectives like Islamicus, or even prepositions like de or super with nominative forms as prevailed in modern Romance.

I suppose it could make sense if you thought about it. However, it would be phrased much better to just write “What is your question about Islam?”