"My little sister is very quiet at school."
After thinking about it, I think the difference is that with "学校に” you're pretty much saying "my little sister is at school being quiet" or "my little sister will be at school being quiet", rather than the intended meaning of "when my little sister is at school, she is quiet"
Yeah, in this case 静か(しずか) would be followed by です rather than います in the same way that you would say 「好き(すき)です」rather than 「好きいます」。
The best way I can describe this in English is that it would sound more like you are saying, "My little sister is very quieting at school." Rather than, "My little sister is very quiet at school."