To establish a second-person point of view in the paragraph, it could be revised to use the pronoun "you" in place of the third-person perspective. Here’s a possible revision:
You stood looking after her with a beaming face, and you began to think that you would like Mrs. Montague, too, if you knew her long enough. Two days later, you were quite sure you would, for you had proof that she really liked you. When her little boy, Charlie, came to your house, he brought something for you wrapped in white paper. Mrs. Morris opened it, and there was a handsome nickel-plated collar, with your name on it—Beautiful Joe.
This version uses "you" and places the reader directly into the narrative.