Firstly check the SQL Logs to ensure a connection is being made from your Sense server, then check your SQL server connection security to ensure the QlikSense services username has access to your SQLServer instance (your DBA should be able to help)
I would guess the issue is at the SQL Sever side, the logs should pinpoint the issue, more often than not it's SQL Server security.
Failing that , then firewalls etc , ports and so on, just throwing ideas about as I dont know your config
What was happening was we had the ODBC drivers installed for the wrong version of SQL Server and not on the QlikSense server. We installed the ODBC SQL Server driver for the correct version of SQL Server and we're able to connect saying we don't have access to the data source, so we'll check the SQL Server logs.
Once we get a resolution, we'll update.
Here's the resolution. In the initial install, the service account was not tied to the services. When installing QlikSense on the server, make sure the QS service account is the account doing the install to make sure all of the services are working properly.
We had to do an uninstall / reinstall of QlikSense and then we were able to connect to the databases.