Up at the Schulman Grove and Patriarch Grove the only green you're going to see is the Bristlecone pine trees themselves. There may be a few small seasonal flowering plants that grow low to the ground, and who's lifecycle is very short in the spring, but don't expect to see them in July. With this year's ongoing drought they may not even bloom this year.
The Grandview campground near the start of the entrance road does have some drought-tolerant pine trees and shrubs around it that'll be green, but that should be the extent of any 'greenery' in the White Mtns. Grandview is a nice place for a car camp as its not heavily used; people who come up to the Bristlecone forest are only there on a day trip. If you want to stay overnight just make sure you bring your own water. Great views over the Sierra Nevada mountains and Owens Valley from the back (west) of the campground, especially at sunrise and sunset.