Closed captioning, in its infancy in the 80s, often dropped words and letters by accident. In the original broadcast of this episode, the captioning of Riva's line, "We could dine together," lost an N, resulting in a rather bizarre exchange. Riva: We could die together. Troi: I'd like that.
Early on in the episode, just after Picard realises what the terrorists are up to. He fights and disables one of the terrorists, a human played by Tim Russ, using what appears to be a Vulcan nerve pinch (although in the script it is referred to as a "carotid artery block"), which he learned from Sarek in an earlier episode. It's ironic that Tim Russ would later go on to play the Vulcan Lt. Tuvok on Voyager, and use that very same move on many an occasion.
During the series, a number of times the shot of the Enterprise moving toward the destination at Impulse power is re-used, and there is a figure moving from left to right along the Conference room windows. According to the LCARSCom.net website, the figure in question is Captain Picard.
In this episode, the stripe on Picard's door is light purple. Usually, the info stripes are brown with white lettering. Also, you see that Picard's quarters are on Deck 9, and has a view of space, whereas Kirk's cabin was further inside, with no windows. This was a means of protecting him from attack.
When Captain Picard is reading Hotel Royale in his ready room, he comments that the book's first line, 'it was a dark and stormy night', is "not a promising beginning". This line is actually the first line of Edward Bulwer-Lytton's Paul Clifford; the line has become so clichéd that it frequently appears in satirical works to denote humorously incompetent or overly melodramatic writing.
Stephen Hawking guest stars in this episode, playing himself as a holodeck character (The "Poker with Einstein" program). This came about when he visited Paramount Pictures to promote "A Brief History of Time", and then told the Paramount people that he had always wanted to visit the Enterprise. He not only got to visit, but the writers added this special scene just so professor Hawking could appear on screen in a Star Trek episode. Hawking also reportedly stopped by the warp engine, smiled and said "I'm working on that".
The current King of Jordan appeared briefly as an officer saluting Picard in one episode.