Hey Theo! Short answer: Nope! The Wi-Fi bill is super boring; it just shows data usage, not your search history. Phew! 
Now, the geeky part: a tech-savvy person could see the websites you visit by checking the router’s logs, but not your specific Google searches.
For true stealth mode (for all that… uh… “homework”), a VPN is your best friend. It basically puts your internet traffic in an invisibility cloak.