Where can I find information on interfacing an MCU with an HDMI display?
As a reference design, check out this link to an Arduino compatible shield and library. There it shows the schematics and what chips they used for the MCU (ARM cortex M0+ and ESP32 MCUs) as well as the display drivers (RA8876 TFT controller and CH7035B HDMI encoder).
There are examples there too to help you figure out where to get started.