There is no way to make money from Discord directly. There is no affiliate program or monetization. This is due in part to the fact that Discord has no ads, which are the go-to for many monetization programs.
That being said, you can utilize Discord as a platform for making money, if you have something to sell. Creating a server around a product or service, or using your existing communities to help promote a new product or service, can result in some extra income from Discord.