Huston's realization of the Pequod was constrained by the materials he had available. The film's Pequod -- nicely observable in this YouTube clip from the movie -- had earlier served as the Hispaniola in the Disney version of Treasure Island and had first seen the light of day in 1887, when it was christened the Ryelands (according to Wikipedia).
As I now understand, "quarterdeck" refers simply to "[t]hat part of the upper or spar-deck which extends between the stern and after-mast, and is used as a promenade by the superior officers or cabin-passengers" (per the OED). It might be raised ... and it might not. If it's raised (and has several other characteristics that I don't fully grasp), it's a generally called a poop. But "poop" might also be used to refer to the stern, period; hence, we sometimes find the phrase "raised poop," which at first glance might appear to be redundant. Then there are "raised quarterdecks," which some say are the same as poops, while others make a distinction.
In any event, I'm using "poop deck" here to mean a quarterdeck that's a few steps higher than the rest of the main deck, like that shown in the John Huston Pequod. For another example, here's a nice photo of the main deck of an 1877 sailing ship, the Elissa, with a poop deck.
Melville, as far as I can tell, does not directly indicate whether the Pequod had a poop deck. From his memorable description in Chapter XVI ("The Ship"), we learn that this "rare old craft" had been in service as a whaler for "more than half a century[.]" Thus, if we assume that Ishmael shipped aboard her in the same year that Melville sailed aboard the Acushnet -- i.e., 1841 -- then she must have been a whaler since before 1791. On top of that, we don't know if she began life as a whaler or was converted from some other purpose. We cannot assume, then, that the Pequod's architecture matched that of vessels built in the 19th century.
Still, for what it may be worth, the whaleship Lagoda, built in 1826 and a half-size model of which forms one of the crown jewels of the New Bedford Whaling Museum, did not have a raised quarterdeck. She started out as a merchant ship and, some 15 years later, was refurbished as a whaler. The companionway leading down to the living quarters below the quarterdeck was protected by a boxy deckhouse on one side of the quarterdeck. (By contrast, in the movie's Pequod, as in most tall ships represented on film, the companionway is through double doors in the face of the quarterdeck [or the forward bulkhead?], like the portal at the top of a cuckoo clock.) The Lagoda also had a skylight running fore and aft next to the deckhouse, protecting the steersman's compass and lighting the aftercabin salon. The Pequod, we learn in Chapter XXXIV ("The Cabin-Table") had a skylight as well, through which one could peep at the captain and mates at dinner.
A Lagoda-like layout is what Mead Schaeffer shows us in one of his colorful illustrations to the 1923 Dodd Mead Moby-Dick. (All the illustrations are beautifully reproduced here; scroll down to the one showing Ahab on the quarterdeck.) But since it's possible that Schaeffer based his illustration on the Lagoda model, which was begun in 1916, his illustration should not be taken as a separate data point.
Melville's reference to a skylight through which one could see the mates dining may suggest that the Pequod did not have a poop deck. In addition, his mention of the Town-Ho's poop in Chapter LIV may indicate that, when a ship has a poop, Melville says so.
Until additional evidence comes in, Melville's silence on the subject is enough for me: I hold that the Pequod did not have a poop deck. In the words of U.S. Sen. James F. Simmons, of Rhode Island, during congressional debates over secession:
[T]he distinguished senator from Texas says that the old men who made this Constitution, of all things in the world, knew nothing about it; that they were very good men for generals, and such like, but they knew nothing about the Constitution. Well, sir, they are good enough authority for me.(Quoted in David P. Currie's The Constitution in Congress: Descent into the Maelstrom, 1829-1861.)