I have a VPS with Windows Server 2012 and with SqlServer already installed. And I developed a Website with a BDD connection on my local PC. The connection string that I use, I generated it with Visual Studio when connecting a table from the same IDE.
In my VPS I can not install other programs , is there a way to get the Connection String, only with the SqlServer or SqlManager?
To obtain the connection string against a database without the need for VS or similar, you can create a file with a .udl extension and open it. This file looks like this: 
In the provider tab you can switch between SQL Server (default), OLE db and in general BDD types of which you have the installed drivers (the OLE are by default in MS)
Once the connection parameters are inserted, click on accept and then right-click on the file and open with notepad (or notepad ++, to your liking) and you will get something similar to this:
I hope it serves you and is what you are looking for.
Greetings !!