Another picture:
http://i22.photobucket.com/albums/b340/vdixit/IMG_0400.jpg http://i22.photobucket.com/albums/b340/vdixit/th_IMG_0400.jpg http://i22.photobucket.com/albums/b340/vdixit/IMG_0399.jpg http://i22.photobucket.com/albums/b340/vdixit/th_IMG_0399.jpg

You said you have 2 Japanese maples, are these pictures just for one of them?

Well if this is only 1 tree the green part is the understock trying to taking over, trace them all the way back to the trunk, you should see that they are from below the graft, prune them off there.

Do it soon, the red part could become weaker or even die when the understock grows for too long.

The pictures are for one of the maple. But the other one has a similar problem. I will try trace the branches and cut them back. Any specific advice on how I should cut them? ANy pitfalls I should be aware of?
Thanks,
Vaishali

This is where opinions differs, if the branches are quite thin, pencil sized or less, you can cut it back flush with the trunk, but not recessed into the trunk.

Use sharp pruners, clean the pruners with alcohol or diluted bleach before you begins.

Don't use anything to dress or seal the wounds they will heal naturally.

If these branches are big and you need a saw, it is best to cut just above the collar. I don't think it is the case unless these branches has been growing for a very long time.