Someone’s bond included me in a no contact order. How do I motion to exlude myself from that so I CAN have contact with the person?

This doesn't sound like a good situation, and I'm absolutely certain that there is a whole lot more to this story.  The missing information starts with what your boyfriend did, that was enough of a problem for the court to grant the no contact order, and would go on from there.

My guess would be that if you are still living with your mother, you have a very steep uphill battle on this one.  In either case, I'm not a Georgia attorney, and this isn't a simple petition by any means.

Violation of a no contact order is often a felony, and they are often interpreted very strictly.  In order for you to have contact with your boyfriend, you need to file a motion to modify the current no contact order.  You can base this motion on the fact that prohibiting you from having contact with your boyfriend, without having been heard on the matter, is a violation of both your first amendment right to free association and your fourteenth amendment right to procedural due process.   If your boyfriend has an attorney, you could contact his attorney who could then take a statement/affidavit from you draft the motion.  However, as you know you must not contact your boyfriend directly.  

If you do not have the ability/access to hire an attorney to file this motion on your behalf, and do not have the legal acumen to draft it on your own, simply expressing your desire to have contact with your boyfriend to the prosecutor's office or a victim's advocate within the prosecutor's office may help influence the prosecutor to modify the order.  However, in my experience prosecutor's often refuse to modify no contact orders even over the expressed desire of the alleged "victim" to have contact with the defendant.

Finally, you should be very careful if you decide to contact the prosecutor on your own, or before you make any statements in open court, due to the fact that you could accidentally say something that could end up incriminating your boyfriend even if your actual intention is to exculpate your boyfriend or have the current no contact order modified.  Therefore, if at all possible you should consult with and/or retain a criminal defense attorney to navigate these rather tricky waters for you.