If it's within 30 days of finalization you can file a motion for reconsideration or a motion for new trial. If it's over a year, there must be a material and substantial change in circumstances to modify. This is fact specific, so contact a family law attorney with your questions.
I agree with Ms. Brochstein's answer, and would like to add that if it has been more than 30 days and there is a significant change in ciircumstances that would necessitate a change in the order, ie. other parent does not work and therefore does not need to claim dependents, you could file for a Modification to the Order. Please see a competent attorney for a consultation to discuss your matter in detail.
Procedurally, the other attorneys are correct. There is another issue though. The IRS, through federal laws and regulations, decides who gets to claim the child. (See IRS Publication 501) A state court judge does not have the authority to order the parties to split the tax deduction in a manner contrary to federal law. Many people put those provisions in their divorce decrees (which I think exposes the obligor's attorney to malpractice claims), but they are only enforceable as a contract, which is to say not enforceable at all in Texas against most litigants.
In other words, you'd probably be wasting your time to try to get a new trial and have this issue litigated.
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