If you are going to play on a LAN (local in the same room or same place) and not online it's very easy. Follow these steps before launching the game:

1. both of you need to turn off your firewall in the Windows Security settings (this is a must) turn it back on when you stop playing.
2. Whomever hosts the map hits Windows+R on their keyboard and type CMD in the box
3. in the the command window that pops up type ipconfig and hit enter.
3.the IPv4 address is your local LAN IP address

Now launch Quake

4.Whomever is hosting the game go to the Multiplayer menu in Quake, follow the steps to host a game (map, game type, etc)
5. when the game is running on the host the other player brings down the console by hitting ~ key. that player types "connect xxx.xxx.xxx.xxx"

the X's are the IP address you discovered earlier. Usually it's like this: 192.168.1.11 or something similar.

that should do it.