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When you have a question like that in narration, you are essentially narrating the protagonist's thoughts. If you put quotes around them, or italicized them, and made them present tense, they would...
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#3: Attribution notice added
Source: https://writers.stackexchange.com/a/11118 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#2: Initial revision
When you have a question like that in narration, you are essentially narrating the protagonist's thoughts. If you put quotes around them, or italicized them, and made them present tense, they would be dialogue. As long as you keep that in mind (and don't overuse the technique, as Watercleave correctly notes), it's perfectly fine to do.