Remembrance - S1-E1
Character mistake: Picard and Data are playing poker at the beginning of the episode. Before the actual faces of the players are revealed, we are shown close-ups of their hands as they put down the chips for the bet. When Picard says his first line in the series it's "See", and it is literally true, because in the previous shot, Data put his 5 cards down on the table and face up. (00:00:50)
Character mistake: Dr. Agnes Jurati, one of the Federation leading scientists, is bored during the hyperspace travel, and so she chats a bit with the captain. While she gives her quirky speech, she casually mentions that "there are over 3 billion stars in our galaxy." She's not wrong, technically, but the number of stars in our galaxy is estimated between 100 and 400 billion. She is way off. (00:08:40)
Suggested correction: Two things. You state that she's not wrong, which she isn't. The fact that she chose an odd turn of phrase doesn't make it a mistake. Plus, you reference the number of 'galaxies' in our galaxy but I am guessing this is just a typo.