To connect to the GIS Database within the ArcGIS Pro software select:

Insert > Connections > New Database Connection. This will open the "Database Connection" dialog. Select "SQL Server" as the Database Platform and type in "gissql" as the instance name. Ensure that the database named "hpgdb" is selected. In most cases you will want to select the Authentication Type for "Operating System Authentication". 

Note: In rare cases you may need to change underlying database schema. To do so 
you will need to change authentication type to "Database authentication" and
supply a specific user name and password to access the database with elevated
privileges. The GIS division within ITS is the keeper of these credentials.

Connection parameters should match those shown here:

For more details regarding database connections within ArcGIS Pro you may consult the following ESRI hosted link:  Connect to Database

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* If when attempting to connect to the Enterprise database you receive one of the following error messages, please refer to the solution below.

Connection Error:   
Connection was attempted with an older version of SQL Server client 
communications software that is not compatible with the SQL Server database 
server.

Connection Error:   
Unable to get database list.

A database connection exception has occurred. Unable to connect to database 
server to retrieve database list; please verify your server name, user name and 
password info and try again.
Connection was attempted with an older version of SQL Server client 
communications software that is not supported.

Solution:

Navigate to your MyESRI page, then select the  My Organizations tab > Downloads tab, and finally Database Support Files (on the left side, under Downloads), as shown below.  Locate the appropriate Microsoft ODBC Driver for SQL Server, download and install on the client PC.  This should resolve the database connection issue.