Several things have to come together to do what you are wanting.
First, as Gonzo stated, you need to get your lens as close to the fencing as possible. Second, it is best if your subject is back away from the fence itself, the further away the better. Third, use as large of an aperture as you can, and finally, the autofocus really depends on your camera and the mesh size of the fence, so use the smallest single point you can and attempt to get it within a void on the fence so that all it sees is the animal, if you cannot, then make switch to manual.
The reason you need all of these to come through is an effect of depth of field. By using the larger aperture on your lens, f2.8 works really well, as it has a shallow depth of field and a very narrow plane that would appear to be in focus, and everything else blurs out. The reason you want your subject back away from the fence is so that the subject and fence do not come into that same depth of field plane, where it becomes obvious.
Remember though, different styles of lens have different depth of field profiles. So the closer you are to the subject, the shallower the depth of field. The further away, the wider the depth of field for a given aperture.
So, when using your D610 and 70-300mm lens, set the aperture at its maximum value (ie. 4.5 to 5.6) for the given focal length and get as close to the fencing as possible. Keep in mind that if you do not use the lens hood, then you may wish to make sure that you have a UV or Haze filter on the front element to prevent a metal fence from scratching your lens.