According to The Japan Times "A Dictionary of Basic Japanese Grammar" textbook, the grammar pattern ～おうと思う can't be used as a question. As an example they state that you can't say あの本を読もうと思いますか。
When the subject is not the first person, [...] the nonpast form of omou cannot be used.
...
The reason why [it] is unacceptable is that omou represents an internal feeling of the speaker alone.
...
It is also to be noted that ~yō to omou cannot be used as a question.
(p.570)
Why is this?
～うと思う can be used as a question, and あの本を読もうと思いますか perfectly makes sense to me.
But it's also a nuanced sentence. This roundabout question is asking the listener's wish or intention very explicitly. I feel this question form is more often used as a rhetorical question (or 反語 in Japanese). あの本を読もうと思いますか sounds to me more like "Do you really/still want to read the book?" rather than simple "Would you like to read the book?"
I happen to have the book (IMHO excellent source by the way), so let me add some context..or well, let me attach a screenshot to be faster:
It seems that the key point here is in the writer stating that the verb 思う
represents an internal feeling of the speaker alone. Therefore, when the subject is the third person, omou has to be replaced by omotte iru ...
It is interesting to read in the answers by some native speakers though, that what is reported as a non-grammatical example, sounds actually OK to them. Maybe this fact that 思う represents an internal feeling a subtle grammar point that often wind up to be neglected? I am sure that even in my native language there would be something I would feel as correct but actually isn't.
I think あの本を読もうと思いますか? make sense. It is translated as "Will you read that book?".
Another example is テニスをしようと思いますか?(Will you play tennis?) but I don't often hear that though.
したい which means "want" may be more common than them like あの本を読みたいですか?(Do you want to read that book?).
For this question, I think a verbatim quote (with context) from the textbook would be really useful in understanding the author's intent.
Edit: Now that it has been provided by Tommy - I guess not really. The author mentions how it would be inappropriate to use it to refer to a third person, but I disagree that it "cannot be used as a question" when referring to the second person. A google search for "うと思いますか" gives quite a few hits (学外実習アンケート集計, 「野菜の摂取や身体活動に関する意識」について (Tokyo Metropolitan Government), 食に関する意識調査), many of which are from questionnaires. The textbook being from 1986 might have something to do with it.
"~うと思うか" does sound like it makes sense, but there are definitely more clear ways of expressing what you probably intend to say when you use that phrasing.
If we take the example "読もうと思うか", are you asking:
To me, "読もうと思うか" sounds like number 3 more than the others.
思う or 思います are usually used only when the subject is the first person. When the subject is not the first person, you have to use 思っている (or a variant of it).
Examples:
彼は正しいと思います。kare wa tadashii to omoimasu.
I think he is correct.
彼は正しいと思っている。kare wa tadashii to omotte iru
He thinks he is correct.
