Yes, you can. And this is the time to do it. Greenwood cuttings work best if you plan to root cuttings. The difference between greenwood and softwood is that greenwood is just starting to firm up and softwood is still soft. Its really hard to tell when that is.
Take a cutting 4 - 5 inches long. Remove all except the top two leaves or so. Treat with rooting hormone and plant.
The easiest way to get new azaleas is to bury I section of stem that is still attached to the plant. Mark the position where the stem touches the ground. Dig a sloping trench about 3 inches deep leading from the stake to the azalea plant. Remove the leaves from the part of the branch that will be lying in the trench, scratch the bottom of the stem near a node and apply some rooting hormone. Peg the branch to the bottom of the trench and replace the soil. Water. It will take a few months but this is the way I have done it in the past. Don't cut the stem from the parent plant until you absolutely know it is rooted.