Before we could attempt to answer your question, we had to figure out what a green giant cypress is. Turns out the "Green Giant" is a trade name for Thuja standishii x plicata. The little "x" in the scientific name is our first clue that this is a cross or hybridization between two species of the genus Thuja and the family Cupressaceae. Most such crosses fall out of the expertise of Smarty Plants, which is the growth, protection and propagation of plants native not only to North America but to the area in which they grow naturally. Crossing two plants always complicates knowing what the characteristics of the resulting plant will be. In our Native Plant Database there is Thuja plicata (Western arborvitae); if you follow the plant link you will see it does not mention any poisonous attributes. Neither this website nor that of Thuja occidentalis (Arborvitae) mention wildlife browsing these trees, which could indicate the spikiness of the plants discouraged browsing, so the cows might not even want it. However, just to make sure, we investigated the "sandishii" part of the Green Giant hybrid. This website from the US National Arboretum also indicates no poisonous parts of the plant, which originated in the Honshu and Shikoku islands of southern Japan.
If you wish to double check on our information, here is a list of websites listing poisonous plants:
Databases to search (by scientific name) for plants poisonous to animals:
FInding the plant listed is a good indication of toxicity. However, not finding the plant listed doesn't guarantee that it is non-toxic, but it increases the probability that it is. It is a good idea to check with your veterinarian.