Can I file a child custody for my son against my ex girlfriend who is an immigrant?

You would not file another case against your ex girlfriend but would need to file a response to her child custody case. (Also, assuming the jurisdiction is correct in that she is in the proper court.) No one can predict what the court will do in your case. The fact that you are heavily involved in your son's life is important and should be brought to the attention of the court through evidence at a hearing and through your response and declarations. Also, since your son is 13, you may want to consider where your son wants to reside. At age 14, the court must listen to the opinion of a child as to where/with which parent they want to reside. However, the court does have the discretion to still listen to a 13 year old child's opinion but it is not mandatory that they do so.