'begging the question' is the name of a specific logical fallacy and is not grammatically correct. You could say 'begs for the question' or as the parent comment suggests...

It's true that people use it that way without issue in everyday conversation, but technically "begging the question" is a logical fallacy in which a statement/proposition presupposes its own truth.

See http://begthequestion.info/ for further info (and a laugh that this site exists).

"Begs the question" refers to a logical fallacy[1], though in modern language usually it's conflated with "invites the question" (which is what people generally mean when they say "begs the question" outside of a comment thread on Reddit).

[1]: https://en.wikipedia.org/wiki/Begging_the_question

I think this article sums it up:

http://www.quickanddirtytips.com/education/grammar/begs-the-...

tldr: Begs the question is a formal term for when a conclusion is not supported by given arguments. Using it the way most people do is technically incorrect, but has become common enough that it is in the gray area where one can consider it the new correct usage.

The transitive form "Begs the question <question>" is long attested, quite common, very clear and readily distinguishable from the intransitive form "Begs the question" [with no direct object], which refers to the petitio principii fallacy, when people prefer a translation into somewhat archaic English to the Latin.

In fact, the attempts I've seen to quantify usages (including those by prescriptivist pedants still trying to pedal the idea that the intransitive usage is the only correct one) find the transitive usage to be the most common, even in publication.

The intransitive use can even be seen as a generalization and rationalization of the transitive use, wherein the transitive use becomes equivalent to the intransitive use with an implied direct object of "the question which the argument was intended to resolve", which (while not the original etymology of the intransitive form) actually makes the intransitive form sensible and has a closer relation to the modern English sense of the words in the phrase than the original etymology of the intransitive usage.

Prescriptivist pedantry on this point is, if this is possible, even more obnoxiously pointless than that directed against the singular usage of "they".