My wife wants me to relinquish my daughter's right if we are getting divorced. She also said that she is not going to ask for any child support and alimony. If we really divorce, by law, Do I have to pay child support, even she is not requesting?

I don't think she can just not request it. It's not for her, it's given for the child, and I don't believe she has the right to just give that up. You both created the child, you both have a responsibility to raise the child and to support the child... with your money, your time, your emotions, etc.

Also, at least in my state, a parent can't relinquish their rights and responsibilities unless there is someone else willing to step in and take them over... such as a stepparent.