The sentence already has a VS inversion: "Do you". It is not correct standard English to add a second VS inversion, such as "are we", although it does sometimes occur in nonstandard native speech. The "we are" is in a subordinate clause, and the subordinate clause is not structured for the interrogative.

No, "Do you know where we are going?" is correct; however, without the "Do you know", "Where are we going?" would be correct.

Thanks, it totally makes sense now. "Where we are going" was an affirmative (sub)sentence because the interrogative is already expressed in the "Do you know" part.