Deductions & credits

Here's my worry, so I'm the Father and the only one with a source of income and can file for a tax return. I have a son, who's 1 now. Sometimes he stays with his mom and sometimes he stays with me.

I planned on filing him on my tax return. However I heard from a relative because she (the mother) receives SNAP (food stamps), that if I do, they will try to put me on child support and fine her for fraud.

Which I don't understand. I'm just trying to get clarification and don't want to get into any trouble.