If you would like to play the game in a LAN environment, simply start the game by using Play button in GOG GALAXY. Here you can host or join LAN games. You will have to type the ip address of the computer you want to join, within a local area network.
If you would like to play the game over the Internet, then you will have to use QuakeWorld (select the game in GOG GALAXY -> More -> QuakeWorld (client)). In order to join the game you will have to know the IP address of a dedicated server from QuakeWorld servers you wish to connect to.
Naturally adding the game installation folder, to your security software exceptions list will be required. If DirectPlay is also not installed, then you will have to install DirectPlay. This article should be helpful: