There are a couple of ways to end up with green hydrangeas in your yard. I say “end up with” because to my knowledge most hydrangea flowers don’t start off being green, but turn green at some point.
The blue flowers on my shrub would naturally turn light green with shades of lavender once the growing season was winding down. This would happen in late August and into September in my area of New Hampshire.
The Annabelle hydrangea flower is white but then turns light green so you will end up with green flowers if you grow that variety.
For bouquets purchased from a florist, green flowers can be achieved by spraying with special coloring to give them a green tint, or added to green colored water so the flower color is changed.