I hope she can but I do not know as this is not an immigration question. As a general matter, child support should have nothing to do with her or child's immigration status. If anything, it is a shame you are on LPR status and your child is undocumented.
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If that is where she resides and obtained the divorce, then yes.
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Yes, she certainly can seek to obtain your financial support for the child through the courts wherever she and the child reside (or the child's home state if different) if you don't provide support voluntarily.
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