Yes if someone has website traffic forced to go through a proxy server when you use their WIFI on their router then they could track your browsing history and see what websites your visiting or what you're downloading.
Most people are usually not that tech savvy or bother to track peoples browsing history on their WIFI though so I wouldn't worry.
When a router is setup there's no default option built in though to see browsing history of the user of the WIFI internet connection.
You'd have to setup some software on the other computer and on your computer to get a complete view of browsing history.