I am digging a 18" diameter hole for a carport footing. There is a 3 1/3 root that runs through the place where the hole is. The hole is about 20 feet from the tree and the root is about 2 feet from the surface. The tree is at least 40 years old and over 40 feet tall. Can I cut this root without doing any damage to the tree.

Hi,
Probably Yes.
Roots to trees extend on out to at least the full tree branch diameter. Cutting one root should not cause any damage to the tree.
I had one cut down last year, big one. I'm still getting small growth from all around where the tree was, cutting them again every time I mow the grass.
Best wishes.