Is there a way to use CB Query to insert into a database?
Nope, it's for pulling data and not inserting data.
If CB Query (or other CB plugins) are not designed to do this, do you know of any other solutions that would do custom table inserts (I saw Alter reports suggested in one of your other posts)
You could use CB Auto Actions to push data in multiple ways. For example you could use a Query action on the after user update trigger to execute a query when the user updates their profile. The Query action can work with internal and external databases as well. There's also the Code action if you need to use PHP. If you need to make an HTTP request you can use the Request action.