Create a custom workout on Garmin Connect with two steps; Interval with duration of 6.22mi distance then CoolDown with duration of "Lap Button Press" if you wish to allow for any extra distance you'll run on a 10k course.
You could also change settings in GC to use km instead of miles then build the workout as a 10km interval before switching units back. I don't think it really matters in the end since 10km is 6.2137mi.
When done, send the workout down to your 610.
Now, with all this said -- I'm not sure you'll end up with the result you want in an actual 10k race simply because people nearly always run somewhat longer than the race distance. Remember the course is measured such that the absolute shortest possible path is the certified distance. Since most people aren't able to make the tightest corners and run perfect tangents between those corners, most people end up running a bit more than the stated race distance.
If you set up your watch to count down 10 kilometers from the start it will probably mark the 10km as finished well short of the real finish line.
