All the World Is Birthday Cake - S2-E5
Corrected entry: Everyone on the planet freaks out because two of the crewmembers have birthdays soon, marking them as "Gilliak." But no-one makes the connection that birthdays, dates, years, etc. will be entirely different on this planet than under Union dating.
All the World Is Birthday Cake - S2-E5
Corrected entry: Why would the doctors on Rigor 2 refer to the process of removing a baby from the womb via surgery as a "C-Section"? Even though the exact etymology of the word is uncertain, it certainly was derived from Earth names and works (Lex Caesarea). There's no way an alien planet would use the same term.
Correction: The doctors on Rigor 2 are not speaking English. What they are saying is being translated into English by the crew's Union translator.
Correction: While I agree the premise is a bit shaky, my understanding from watching the episode is that the people of the planet had it so ingrained in their culture, they weren't willing to consider that. Regardless of where someone was born, if they were born during the period of that sign on their planet, they were under that sign. While watching the episode, I hoped Captain Mercer would use your logic to free them, though. He could have simply said "You know what? We have that same sign in our culture, but because the stars are different for us it comes at a different time."