I’m tempted to just say”no” [but that would be too simple].
The cultivar Thuja 'Green Giant' is a hybrid between T. standishi and T. plicata and certainly could not be considered a native species here in Texas. Near relatives in the US are Thuja occidentalis (Arborvitae), which is native in the Midwest to Northeast, or Thuja plicata (Western arborvitae) (one parent of the hybrid), which is native to the West Coast.
There are several good options for privacy plants for Austin. It just matters, in relation to the Green Giant, which characteristics you might accept being a bit different.
Here is a good baseline of advice about privacy plants for the Austin area: [I found these by searching the Mr Smarty Plants archives for “Privacy” – There is a LOT of material there]
Fast growing native evergreens
Fast growing shrub and explanation of searching procedure on the “Suggested Species” database
And, since you mentioned “tall” here are three responses directly about trees:
Trees for privacy screen in Austin
Tree landscaping in Austin
Fenceline trees for NW Austin
So, what is it about the Thuya Green Giant that you value? When I look at that tree, I see a tall evergreen of slender aspect, which is reputed to grow fast.
Probably the closest local tree is Juniperus virginiana (Eastern red cedar), it normally tops out in the 30-40 ft. range but has a nice “Christmas tree” aspect. Both it and the Green Giant are evergreen and have cedar/juniper type foliage. A bit more rounded and seldom reaching 30 feet tall is our own Juniperus ashei (Ashe juniper); my cedar fever prevents me from being too enthusiastic on that one!
Other good choices for trees for privacy shielding include Prunus caroliniana (Cherry laurel) and the taller, but deciduous Platanus occidentalis (American sycamore), Quercus macrocarpa (Bur oak), and Catalpa speciosa (Northern catalpa) (fast growing)