How many dialogue do we have here?
How many dialogue we have here?
As far as I'm concerned in the answer we didn't have subject we should use auxiliary here.
Your guess is correct. We need the do in the sentence to make a valid question form.
But since we are asking How many, we will use the plural form of the noun dialogues.
How many dialogues do we have here?
The grammatical concept is often described under the term do-support.