Learn Spanish (Latin America): Quiero hacer una pregunta.

In Spanish, you don't usually "ask" a question—you "make" one! Saying preguntar una pregunta sounds very repetitive to a Spanish speaker. Instead, the standard, natural phrase is always hacer una pregunta.

How would I change this to say "I want to ask YOU a question"?

You need to add the indirect object pronoun te (to you). You can attach it directly to the end of the infinitive (unconjugated) verb: Quiero hacerte una pregunta. You can also put it right before the conjugated verb: Te quiero hacer una pregunta. Both options mean the exact same thing.

Could I just use the verb preguntar on its own without the noun pregunta?

Yes! If you drop the noun pregunta (question), you can simply use the verb preguntar (to ask). For example, it is very common to say Quiero preguntar algo (I want to ask something).

Since our current lesson is about reported speech, how would I report that my friend said this?

You would use a reporting verb like dice (says) followed by que (that). For example: Mi amigo dice que quiere hacer una pregunta (My friend says that he wants to ask a question). Notice how quiero (I want) shifts to quiere (he wants) to match the new subject, but the rest of the phrase stays exactly the same.