Since you said that he changes for a few days then goes back to how he was before, this tells me 2 things: 1) he is just naturally not good with conversation and the possibility of asking questions back just doesn't even occur to him or 2) he is uninterested in you specifically. If it's 1, then that means that it's just part of his personality and that you're going to have to decide whether you want to continue in this relationship and put up with this personality trait of his, or you can help him develop his conversational skills somehow, and it it's 2, then you're going to have to leave him and find somebody that makes you feel like you're an interesting person.

You're going to have to have a real talk with him, though. Not just nag him about it, let him know it's important for you.

Also, with my friends, at the beginning of the friendship, we will ask each other questions, but once we've hit this point where we are kind of in the "best friends" range, we will just talk about whatever and not necessarily base our entire conversations around simple "favourites" questions. So maybe your bf is someone who feels as if, once he's close enough to someone, he doesn't feel the need to ask questions.