I know its a pain in the butt but read through all the responses in this thread. You will find several other causes that are not marked as the answer. If everyone keeps repeating the same responses, the thread will get even harder to read through.
Causes have been found to be One Drive, other third-party software
like Mozy, Carbonite, or SugarSync. Dropbox can also put green check marks on your desktop icons. Verizon cloud will put green checkmarks on desktop icons.
For me, it started with the forced update from Win10 version 1703 to Win10 version 1709. Due to other issues, I had to wipe my hard drive and do a clean install of 1709
from an installation disk (ISO). The defaults of that installation were set to automatically sync the computer to other computers and to auto backup several items including my Desktop. I was never given an option during install to set any of these things.
See my post from December 2017 on page 12 of this thread, for my solution to my problem. There is a multitude of causes for green checkmarks to just start showing up on your icons. Only the original poster and the site
Moderators can mark another post as the answer to the original question. And I believe there can only be three answers marked at any one time in a thread. That's why you have to read the whole thread. The site supervisors could
fix this most likely, but they haven't for as long as I've been around.
The Win10 developers could also fix this but they actually think this is an important feature and it makes Microsoft tons of money. Just do a google search for One Cloud. Anyway, I hope you find the cause of
your issue. It should be easy to correct after you discover the reason for the green checkmarks in your case.
And finally, my apologies to the OP for making the thread even longer by replying to a different poster. I know it's against the forum rules, but It needed to be said.