this may be too late, but there seems to be a better answer for this question.
I was searching for the same thing; how do I view the connection string to a particular SQL Server database. I read you cannot do it within SQL Server Management Studio, but you can retrieve the connection string Visual Studio uses, as suggested by another user.
In Visual Studio, go to Server Explorer and choose to create a new connection by right clicking on Data Connections and browsing to the SQL Server to which you wish to connect. I am connecting to both SQL Server and MySQL so I would note this down for myself and others who might be in the same boat, so to speak.
When choosing a new Data Source, specify Microsoft SQL Server for your SQL connection, but MySQL needs a connector. You will need to have a MySQL/Oracle account to download the connector for MySQL.
Once this is installed, you will have an option of Oracle database when you are creating a new data connection for MysQL. You can now configure and connect to your data source and view the connection string for each data source. To do so, right click on the new Data Connection and select Properties. In the Connection String, you can copy and paste this for your Visual Studio connection string.
Again, since this came back in my search results, I am adding this note for myself in the future and possible others may find this useful.