Do I need an SSL certificate when transferring from server to server?

Question

Digital certificates

Hello.

There are 2 servers - one is a web server, the second is a game server.
I need to collect data from a game server and store it in a database. If transmitting data via sockets (SSL protocol), do I need an SSL certificate for this?

I am also interested in how safe it is to simply encrypt the socket with one of the cryptographic methods (the key is known in advance on both servers, i.e. it is not transmitted), and transfer it?

Purpose: Protection against data spoofing.