There are various open source FPGA projects that do HDMI output. Some of them probably run on hardware with enough extra LVDS lanes to also do PCIE. The MKR V

There are various open source FPGA projects that do HDMI output. Some of them probably run on hardware with enough extra LVDS lanes to also do PCIE. The MKR VIDOR 4000 has connectors for both HDMI and Mini-PCIe, and the datasheet for the FPGA on it suggests it might have enough lvds lanes to drive both simultaneously. The fpga4fun.com website steps through how to create HDMI output code, how to make PCI code, and how to use the Xilinx wizard for PCIe support. In past projects that actually tried to make desktop graphics cards, it sounded like the hard parts weren't making video output or talking to the host computer, the hard parts were providing VGA bios support for booting.