Camera not coming on. Battery fully charged but shows low battery

For starters, it's still possible that the problem lies with the batteries rather than the camera. More precisely, with the charger. The first thing I'd do when faced with a situation like this, is to grab my multimeter and check if the batteries are indeed charged. The connector of the D610's battery pack makes this a little bit awkward (there are 5 recessed contacts and there's no way to tell which 2 out of those 5 are the battery terminals proper) but probing all possible combinations should result in at least one pair of contacts having a voltage of at least 7V between them, closer to 8V if the battery is fully charged. If no such pair is found, the battery is almost certainly discharged or defective. Since it is not very likely that two batteries fail in the same way at the same time, the problem might be with the charger as well.

If you can establish, via the above procedure or otherwise, that the batteries are indeed good and fully charged and the camera still says low battery, the problem is likely with the camera's power supply circuit. Since your D610 is less than a year old, it should still be under warranty and my call would be to let Nikon take care of this.