The Air Force Captain that interrogates the aliens is wearing his insignia bars left-to-right rather than the correct up-and-down.
When the MPs first enter with Carlson they are holding their pistols at "Hollywood ready," which is to say right next to their faces. In reality this is the opposite position than the one that would be used. In a real low ready position, the pistols would be held at about lower chest height, that way if they needed to go to a high ready position (preparing to fire) they would simply need to raise their arms and acquire a proper sight picture. Coming from the position they were in, they would have to both lower the pistol and position their arms into a firing stance, as well as pick up the sight picture, it is much too slow, which is why no real training doctrine uses it. "Hollywood ready" exists exclusively to show off the pistol to the camera, a trained MP would never use that position.
The military doctors inject Quark with Sodium Pentothal to try to get information out of him. The idea of Sodium Pentothal as a "truth serum" is a Hollywood myth. Sodium Pentothal is a fast-acting barbiturate. Additionally, the risk of overdose is fairly high, so it would have to be injected much more slowly than depicted here, and would not be injected repeatedly as Nurse Garland does to Quark.
When Quark is speaking to Denning, the lower edges of his facial appliances are visible.
In Quark's ship just before flying into the atomic bomb, Rom's nose appliance is squished and crooked in several shots.
The first US nuclear weapons test at the Nevada Test Site did not occur until 27 January 1951, three and a half years after the Roswell Incident. In fact, the US did not conduct any nuclear tests in 1947.
The text "BLDG A" and "RESTRICTED AREA" seen in 1947 are written in the Chicago typeface. This typeface wasn't invented until the 1980s.
It is said that Quark had to sell his shuttle for salvage to afford passage back to Deep Space Nine. They were on Earth and Deep Space Nine is a Federation outpost. Ships and shuttles are shown to make regular trips between the two places, and the Federation doesn't use money. There is no reason that Quark would have to pay for passage back to the station.
When trying to negotiate with General Denning, Quark expresses interest in gold as currency. To the Ferengi, gold is worthless and is merely a container for latinum (which is a liquid).
Quark says that he'll give the Ferengi warp drive in 1947, centuries before the Vulcans develop it. But it's been established that the Vulcans had warp technology since at least the 19th century, if not long before.