Okay here is my take, and it may sound rude, but my opinions just come off as rude 99.9% of the time.
Green horses will commonly buck in the canter. My green 4 yr old started cantering fine one day, and the next he was a bucking disaster. Cue for canter? He's off on an uncontrollable bucking spree. Stop him and cue again? More bucking. It is NOT an easy fix, but it is possible. You NEED to get after her as soon as she let's out one buck. I go with tight, uncomfortable circles that change direction often. (Most usually at the trot).
I don't know how you ride, but I'm trying to think what would make her buck, being a green horse. With my green horse, if the reins are tight when cantering, he bucks since he is confused. You could be putting a "wall" in front of her, and she could be getting mixed signals as what to do.
Another thing is, if you get off of her after she bucks, or stop cantering her for the rest of the ride, she's learning. She's learning if she bucks, she gets rest, when it needs to be the complete opposite.
One other thing she could be bucking for is that it is an old habit. She probably knows you are scared of her, and bucking gets her a break. If you have a really solid seated friend that knows how to ride horses that buck, ask him/her to ride her one really good time. I fixed a bucking horse this way. My neighbors horse would through bucks a lot, and being the stupid person I am, I would get on him just to ride through the bucks and then discipline him, just so he knows he won't win with everyone. Now, a little girl rides him around, and he is perfect for her. Some horses need to be put in their place by a very experienced rider that can stay on top, and ride it out.
If I was in your position I would start asking myself questions. Am I the cause of her bucking? Is she mentally ready for cantering? Is she uncomfortable in any way? Will I be able to get through this? Do I feel safe, to a degree, to keep riding her?
Personally, if I was you, despite my disliking of giving up on horses, get rid of her and get a good beginner horse. You are in over your head if you feel you can't fix the problem, and if you feel in danger. Confidence will get you farthest with a bucker, and if you aren't confident that there is a light at the end of the tunnel, you don't have the horse for you. This horse could be fixed, but I don't believe with a beginner rider.
But just a tip of advice, you will never avoid falling off horses. Even the best of the best fall. If you end up getting rid of every horse that makes you fall off, then you probably won't have horses for all that long. I have fallen off almost every single horse I have ridden. It is something you really can't avoid.
Good luck with the mare.