Yes, QIE supports database connections using a Named Instance.
To specify a database connection using a named instance you include both the server name and the named instance in the hostname field separated by a back slash.
e.g. localhost\namedInstance
References:
https://docs.microsoft.com/en-us/sql/connect/jdbc/building-the-connection-url?view=sql-server-2017
https://confluence.atlassian.com/stashkb/connecting-to-named-instances-in-sql-server-from-stash-443548795.html