yes...you can prune it and the cuttings will root for you.
Cut the cuttings with a clean pair of scissors or knife.
Remove any lower leaves. Anything that will be under the soil.
You can easily root Begonia cuttings in a store bought medium such as ProMix. If you have potting soil you can lighten it up with perlite. Use damp, not wet, soil.
Depending upon your location you can make a tent for your cuttings, to keep in humidity, from a plastic freezer bag. Seems you are in Georgia? you probably have enough natural humidity that you won't need to tent the cuttings.
Keep the cuttings in light but no sun.
The cuttings should root in a couple of weeks.
When you notice new growth you can pot the cuttings up into their permanent pots.
