The answer is yes you can. There are some different steps that you might need to take. You would need to file a Complaint to Seek Custody and Child Support.  If you are not able to locate where the other parent is, you can file something in court asking for an alternate way to have that parent served; in which it would consist of doing a legal summons by publication. You have to pay for the Summons by Publication unless you are indigent and then sometimes the state would pay.

After you get it published in a newspaper, you get a sheet from the newspaper that you have to send back to the court showing that the publication had occurred.  The court then will give you a court date and giving the other parent some time to respond.  Although, in my practice it has been less often that a parent responds if they are not able to be located.

Even if you get a Pre-trial date that is scheduled, you can still do a Pre-trial Memorandum and things of that nature.  You can also do a Proposed Judgment where you can ask for sole legal and physical custody and having any parenting time or visitation at your discretion.   You can also request for child support as well.