As https://trino.io/docs/current/client.html calls out, there is not an officially supported REST API to execute queries against. Of course, this is interesting as the official clients (like ODBC, JDBC, Python, etc) in fact use the "underlying" REST API to do that very thing. Some insight to what's happening, including with the new spooling protocol, can be found at https://www.starburst.io/blog/trino-spooling-protocol/.

Commercial vendor Starburst (disclaimer: I'm a dev advocate there) does has REST APIs for the management of the cluster itself as detailed at https://galaxy.starburst.io/public-api, https://docs.starburst.io/starburst-galaxy/developer-tools/api/, and https://docs.starburst.io/latest/api/index.html.

If you really want to work directly against the REST API to execute your queries, https://trino.io/docs/current/develop/client-protocol.html is the most definitive documentation. Couple that with the Trino spooling protocol blog post I put in the first paragraph to try to build the mental picture of what's happening.