This is my first time having my peony trees bloom. I was wondering after the blooms die off can I then cut the old bloom off and plant it for a new tree???
No. You can allow the seed pods to ripen. When the pod is ripe it will split open exposing the seeds. You should immediately plant the seeds. Do a search in this forum on seed planting.
If a part of the plant would happen to break sometimes, if you are lucky, you can push the branch into moist soil and it may root.