The Drivers are in the Kernel. So, You don't have to install additional Drivers. Theoretically, You could but maybe You could try first install a newer Kernel. That could offer better Hardware support.
You could take the Liquorix Kernel. But before You install it, make sure that in Your BIOS Secure Boot is disabled because the Kernel only works with disabled Secure Boot. How to install the Kernel:
Open the Terminal and type
sudo add-apt-repository ppa:damentz/liquorix
When added the PPA type
sudo apt update
Then install the Kernel with
sudo apt install linux-headers-liquorix-amd64 linux-image-liquorix-amd64
When installed, reboot and on the GRUB Menu choose ''Advanced Options'' and there the Liquorix Kernel Option.
To get the GRUB Menu if You shouldn't have one, press on Start-Up Esc or Shift Key. When that shouldn't work, You can activate the GRUB Menu manually. To do that open the Terminal in Zorin and type
sudo nano /etc/default/grub
Ithen You can work on the GRUB Configuration. there you should see a white Textblock:
Here, you have to change the Lines 2 and 3 so that they look like this:
GRUB_TIMEOUT_STYLE=menu
GRUB_TIMEOUT=20
The 20 makes that the GRUB Menu appears 20 Seconds. If that should be too long or too short for You, You can set another Value like 10, 30 or 50 if You wish. Or You can set it to -1 and then the GRUB Menu stays until You make an Input.
When You have make the Changes press ctrl+o to save, Enter to confirm and ctrl+x to exit. Back in the normal Terminal Overview type sudo update-grub to make the Changes affect. Don't forget that! Without it, it will not work.
I could imagine that the Adapter might not be liked. Can You use another Connection Method? DisplayPort maybe?