Is it wrong to start your abstract with a question?

As an author, you can choose different styles for abstracts. As far as I know, the most interesting and shortest abstract ever was written in this paper, Can apparent superluminal neutrino speeds be explained as a quantum weak measurement? by M V Berry, N Brunner, S Popescu and P Shukla.

Abstract

Probably not.

As others said, this is a matter of style of convention and taste.

Compare these two abstracts:

Does every compact Hausdorff space admit a compatible metric? In this work we show that the answer is positive exactly in the case where the space is second-countable.

And

We show that a compact Hausdorff space admits a compatible metric if and only if it is second-countable.

Both have the same content, but only one of them feels like it actually invites you to read the paper. It starts by asking you a question, which to some extent is intriguing, and then provides you with a complete answer.

Papers should be something that is read by people. As such, the writing style should not be dry. I'm not saying that you should go overboard with elaborate writing and storytelling devices, but sprucing up your writing a little bit using questions or explanations is a good thing; it can help to make your paper much more palatable.

As for your friend's comment? Well, if he can judge the content of a paper by the first sentence of the abstract, I'm sure that he can skip the abstract altogether and just judge a paper by the title. I mean, why waste time reading two sentences?

"I will never start an abstract with questions, and I will never read this paper if it starts with questions."

These are your friend's personal opinions regarding abstracts.

If you take a moment to read a cursory sample of published research abstracts, how many of them have a question? Isn't the purpose of research to answer questions?

As for your friend, politely agree to disagree. You decide what will be published.

Focus on whether or not your abstract effectively summarizes your body of research and its greater importance to the field. This is the purpose of the abstract, whether or not you choose to use a question or not is a matter of style and the input from your co-authors, editors, and confidants.

Remember that this is your work, not theirs.