If there's a legal services organization in your area, and your income is low enough, you might be able to get a free lawyer.  This won't be a court appointed attorney the same way as in a criminal case, and you don't have any legal right to have a free attorney in a family court case of this type.

The right to a court-appointed lawyer comes into play when a person is facing serious criminal charges, or the equivalent in terms of the possible consequences.  A possible sentence of more than six months in jail is the basic standard, under the landmark case of Gideon v. Wainwright.