the baby would most likely look like the emperor (like most "pregnant with the tyrant's child" stories) plus, this is the medieval times. To be pregnant without being married was considered embarrassing and shameful back then. The emperor can sleep around because, he's an emperor and him having concubines is considered normal. ٩(๑❛ᴗ❛๑)۶
She could quit by telling them that she's pregnant then run away to another country or something
That'll be suspicious since it is now quite known to all of the people in the kingdom that the emperor is searching for a girl the he slept with. Since the people still hasnt known who this girl is and why is the emperor searching for her, they can only theorize or suspect that, that girl is pregnant with the emperor's child.
If ever she said that she is pregnant when asking for a retirement, she would get caught by rumours (some people can catch on fast on these kind of topics ya know like how she suddenly ask for a retirement after the emperor announce to find a specific girl) and also, what IF the emperor ask her to stay (because she's a knight turbalosh and she cant deny the emperor's orders)
So yeah. She picked the best option. Running away
Hope this clears up your mind^^
Why can't you just tell that your pregnant? It's not like they have the ability to identify Gene's.