When you say counter claim do you mean request that he pay you child support? If he files a motion, then you can file a response and motion requesting that he pay you child support. I'm licensed in California. You really ought to consult with a lawyer licensed in your state as to how to proceed.
While trying to avoid paying for attorneys and avoiding the court system seems appealing it can cause a lot of headaches down the road. It is not clear to me the nature of your former relationship so let me cover a few different scenarios.
To file a claim for child support you would typically do that in a divorce decree or a decree of paternity. If you already have a final divorce decree that doesn't deal with the issue of child support, which seems highly unlikely, you would need to file a petition to modify the decree. If you were never married to your ex and never got a final divorce decree you should file for divorce and have the matter finalized.
If you want to continue to avoid going to court you could always find a child support calculator plus in the numbers for you and your wife and show her how much support you should each be paying. There is a link on my website http://www.ipsonlaw.com/utah-child-support.htm. I don't recommend this option but the choice is yours. Good Luck.
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