Green flowers are surprisingly not all that uncommon in nature. At least here in the southeast, we have many species that produce lime-green flowers.
One of my favorite fragrant shrubs for our climate is the sweetshrub or Calycanthus floridus. While typically a deep red-purple to almost black in the wild, there are populations with greenish-yellow flowers. 'Athens' is the only green-flowered cultivar widely available that I'm aware of. The fragrance, which you just have to experience, suggests slightly overripe fruit, some say strawberry, but I smell something more alcoholic, like a warm sangria or fruit cocktail left out in the sun. As with all fragrant plants, warm temperatures make for the best-smelling conditions- warm still evenings are ideal!
Wild populations display a great range and variability of fragrance- from almost nothing to wine or apple cider vinegar. So it is important to always buy a Calycanthus in flower or of a reliably fragrant variety. 'Hartlage Wine', one of the newish hybrids even smells like freshly ground black pepper!
Another great shrub or small tree found naturally throughout the Piedmont is the green buckeye or Aesculus sylvatica. Buckeyes grow in mesic hardwood forests, often in great abundance. Green buckeye intergrades with the red-flowered A. pavia in the western parts of Georgia. The resulting mix results in a hybrid swarm of light pastel peaches to rich salmon reds. These shades are common west of Atlanta.
But I find the lime-green ones the most attractive. Perhaps because that's the color I grew up with around the woods of Jasper County. This plant, to my knowledge, isn't fragrant, though, come to think of it, I've never really made an effort to sniff it! It does seem odd that it wouldn't have at least some fragrance... I assume the same kinds of flies and beetles that pollinate sweetshrub would also pollinate it.
My third green flower is an herbaceous wildflower. Trillium cuneatum, or toadshade, like the sweetshrub, is mostly a deep purple-red flower. But in many places throughout the southeast, significant numbers of green/yellow specimens can be observed. And just like the sweetshrub, these often emit an aroma of fermenting fruit. Trilliums, contrary to popular belief, aren't hard to cultivate in gardens. All they require are decent well-drained soils in typical woodland or hardwood shade conditions. Their only real nemesis are slugs and deer. Other than that, they thrive with very little care, even neglect.
Trilliums are so diverse and variable. Each individual has its own character, with its own unique leaf mottling (pattern) and subtle flower color. And there are just so many species, many of which produce green to yellow flowers and all sorts of distinctive fragrances. Just be sure to buy your trilliums from a reputable source as many of the cheaper rhizomes sold on eBay and Etsy are most likely wild collected. If you aren't paying at least $20/ rhizome, it probably isn't legit. They are expensive because they take a long time to propagate. And if you are in the deep south, I'd suggest sticking with the sessile species also from the deep south. Many pedicellate trilliums just aren't happy in our climate.