My oldest brother got into a relationship with a young woman with two children from a prior relationship. He and this woman became pregnant with a little girl. They ended their relationship, and since my niece has been born (almost two years now) no one in this family has seen her more than a handful of times. My brother pays child support, and every time my brother calls and asks to see his daughter or take her overnight and spend some time with her, she denies it even though they have joint custody. We finally have a court date set for June 2nd and are trying to serve her the papers to show up in court, however she recently moved and is not giving us a current address to send them. She is aware that my brother filed a court date because they spoke on the phone and she still is not giving an address for us to send them so what can we do now? Do we still show up to court and hope she'll be there? Do we call and cancel the court date?

If he is paying through an agency then you could get a court order for them to either compell or to deliver the documents. Also you can send service by mail. You send it to the last known address. You can also publish it in a newspaper under the legal notices section. Where is he sending his support payments to?

His support checks come directly out of his weekly paychecks from work. So if he posts it in the paper under the legal notices section is that all he would have to do or would he also have to send the papers by mail?

He needs to try everything in his power to contact the other party. Depending on your court system there are very specific rules regarding notice. If it is coming out of his check then most likely it is going through a support agency. They should have an address. Anything you send should be registered mail. And if the checks are going to her mothers house according to agency addresses then send copies there also. Cover all your bases.
If for some reason and Im sure it will happen that the agency refuses to cooperate then he may need to file a exparte motion with the courts for them to "compel" and force the address from them.