Whether or not they have Medicaid has nothing to do with you being able to claim him as a dependent. If you qualify to claim him, you are able to.
However, if you claim your son as a dependent, it will likely disqualify your girlfriend for Medicaid.
Also, although rules vary by state, your son's eligibility for Medicaid is likely based on your combined incomes (because you all live together). So you need to let the Medicaid office know the entire situation.
