The "It is green" line Data says is a throwback to the same line Scotty says in By Any Other Name (1968) when describing a bottle of alcohol he had found during a drinking binge (to stall an enemy invader) and didn't know what kind of spirit it was.
A close-up of Scotty's hand on a control panel was actually a shot of another person's hand. Actor James Doohan, who plays Scotty, lost his right middle finger during the World War II D-Day invasion of Normandy. (Several quick shots throughout Star Trek (1966) and the first six Trek films reveal this.)
A Dyson's Sphere is a real theory postulated by Freeman Dyson in 1959. But the actual sphere that Dyson theorized was not a solid object as seen in Relics. In an interview for the website Meaning of Life TV, Dyson himself was asked if he had known that a Dyson Sphere was featured on the show and mentioned specifically by name, to which he replied he was aware of it because his daughter had sent him a taped copy of this episode. He went on to say that the episode was "fun to watch, and even though it was all nonsense, it was still quite a good piece of cinema."
Scotty scolds Geordi for being honest with the Captain and telling him exactly what to expect instead of making him expecting less and then offering more. Indeed, in Star Trek III: The Search for Spock (1984) he hints to Kirk that he might overstate some problems to make a better name for himself after he solves them.
The scene of the original Enterprise was made by looping footage of the empty bridge seen in This Side of Paradise (1967), and bluescreening it behind James Doohan. Because of this, only small parts of the set had to be built. Ironically, Doohan didn't appear in the episode the shot was taken from. The shot of the original viewscreen was recycled from The Mark of Gideon (1969).