You can claim a "close relative" (child, grandchild, niece or nephew, and a couple others) as a qualifying child dependent if they lived in your home more than half the year (or more than half the number of days since they were born). If their parent could also claim the child as a dependent (such as, you all lived together for all or part of the year) then the parent gets first choice, and if the parent agrees to let you claim the child, your income must be higher than the parent's. The child must also have their social security number before you file your return.
In Turbotax, select "the child lived with me all year" for a child that was born in the middle of the year and lived with you since.
We can't tell you if that will affect the child's eligibility for government benefits, that is a question for Medicaid or for an attorney or financial planner.
