How do I fix this drill hole on countertop?
That's a tough one because plastic laminate is applied to the particle board substrate with contact cement, which is soft and rubbery. It's very good for large areas, but lousy for little fixes like that. To make matters worse, you need to remove the existing adhesive before carrying on.

If you can indeed get a repair disc from a hidden area, here's what I'd do:


Clean out this hole by drilling deeper... just enough to completely remove the old adhesive. You could also scrape it away with a square-ended hobby knife or whatever. 
Remove the adhesive from your repair disc. The simplest way might be to rub the disc on a piece of sandpaper until it's clean. 
Test fit the disc. Figure out the best orientation with respect to pattern-matching.
Set the disc with clear two-part epoxy or wood glue. The key here is going to be to get the patch on plane. See the glare in your photo? If you don't get the patch on plane it'll reflect differently and appear much darker or lighter than the surrounding area. Set it in the adhesive and periodically adjust it as the adhesive sets using a toothpick or hobby knife.
After the initial set has cured and the disc is stable, fill the circular joint with more epoxy or glue to seal it up and improve the appearance. Be prepared to wipe excess off the surface before it cures. 
Once that's done, buff the area lightly with something suitable (like steel wool) to knock off any bumps or edges and make the sheen uniform. Be very careful here to not leave a large marred area. Test somewhere hidden and inspect your work often. This step could ruin the outcome.