Starlink by virtue of being your ISP would have access to any DNS queries you send over the Internet over UDP port 53 in plain text. Starlink is also able to redirect those queries to their own servers. Even if you manually specify 8.8.8.8 or 1.1.1.1 Starlink can redirect traffic to their own DNS servers and return responses as if they came from those servers.

By itself DNS can tell a pretty detailed picture about you and what you do on the Internet without the need for SSL inspection or other deep packet inspection techniques.