The 2nd one, 肃静 is usually used in a very very official situation, like in court, or raising our national flag. 请保持安静 means please be quiet, can be used in a library and in class/ meeting...

I would say use "请保持安静", as it is more common in library or like you said anywhere else where you want to keep quiet

Jack nailed it. It's in the comments. Definitely go with "请保持安静" in your scenario.