In the end, a payment integration is just like any other integration.
You do have your app and an other endpoint which need to talk to each other.
The question each time is; which language do they speak with each other, what are the prerequisites and what is the process of communication between these 2.
So if you want to have a payment integration
If you want to know how to integrate in Mendix then start with this learning path: https://academy.mendix.com/link/paths/44/Importing-and-Exporting-Your-Data
There is 2 free payment integration module is available in the mendix marketplace. You can use these modules for payment integration