You sound like you have a good heart and care about people. I hope you will take the time to go over and introduce yourself to the mom next door. After all, you are neighbors - why not be friendly!

Perhaps during the time you talk, you can tell her about yourself and ask if she would be interested in having coffee breaks together. She is a new mom and trust me, it isn't easy! Ask her questions. It could be that she might be in a situation where she doesn't have close friends or family nearby to give her a break. If that is the case, you have a chance to offer to help her out.