If I send my daughter to stay with her for a year, can she make me pay child support?

Yes.

Generally, if you are going to make a mutually agreed upon change to your existing custody arrangement, especially if it is set forth in a court order, judgment, and/or written agreement, all of the terms of your mutually agreed upon changes should be reflected in a similar written agreement, court order and/or judgment. You should generally be able to calculate what your monthly child support obligation would be to her during this temporary change, and then provide for the reduction in the arrears balance that she owes you in that same monthly amount for each month that your child is residing with her, so that no money actually changes hands. Consult with an attorney to discuss the specifics of your situation and your options.

It seems to me that with an arrearage like this over her head, you should not have to pay child support to her.