You need to file a motion with the court to request attorney’s fees from the other party.
Family court is not criminal court. It is a civil issue. You are only get a public defender in criminal cases.
If this is a civil matter there is no access to public defenders, they only serve on criminal cases. If you're divorcing you can include the request that your spouse pay for your lawyer. If you're suing your spouse civilly you can include language that he/she has to pay in the case they lose. But assuming you're married and facing some civil case that doesn't qualify for a PD, no judge is going to force your spouse to pay your legal costs.
I assume we're talking about divorce. You get a lawyer, and then the lawyer knows how to fill out paper work for that. The lawyer can also tell you if it's possible.