A ‘headless’ machine is just a machine with no keyboard, mouse or monitor attached, which you access over the network only.
I’ve never heard of Windows refusing to boot because of no monitor being attached. But there’s at least one other ‘catch’ that could be the problem.
What‘s common is that the BIOS may fail to boot if a keyboard isn’t attached. (Oddly enough, displaying “Press F1 to continue” – how you’re supposed to do that when it knows you have no keyboard is beyond me.) And it’s easy to miss that error if the monitor is not attached.
On more recent machines and recent BIOS’s there’s often a setting in the BIOS configuration that will allow you to instruct the BIOS to ignore a keyboard not present error. Look for that and set it appropriately.
On one of my older machines that’s not an option, and I actually ended up connecting an old keyboard just to keep it happy and make it work.
But mouse and monitor were not required.