Can I create custom indexes on tables used with Heroku Connect?
You want to create database indexes on tables that are managed by Heroku Connect.
Heroku Connect offers the ability to add a simple index through a mapping's edit page in the Heroku Connect dashboard. While these are only simple indexes, they comes with the additional benefit of being included when a connection's configuration is exported.
If a more complex index is desired, you may add it directly using Postgres but it won't be maintain when you export and reimported to another Connect instances. You will need to manually recreate it.