A general rule that is not always fleshed out well in the language is that you should bond anything that is likely to be energized. If this fence is not near any energized conductors, it probably does not need to be bonded.
As an aside, it is probably fairly well "grounded" by the posts in the ground but that is not the same as bonded.
Places where I see fences bonded is where there is significant radiated power. This can be a radio/TV transmitter or just a big distribution transformer farm.
680 will also require that a fence be bonded if it is <5' from the water.
