Perhaps I can help. To answer your question, no
. You can not file a petition for visitation by drafting a letter. However, you can
file a petition for visitation. The petition has to be drafted a certain way and filed with the court. You can contact the clerk of court to review similar petitions that have already been filed and determine whether they have a blank "form" petition for you to use. You will probably have to hire an attorney to assist you if you are unable to locate or draft a form on your own. Once you have the petition drafted and filed, it will have to be served. There are specific procedures for service as well, and these have to be followed if you are to ever get your case heard before the court. For this reason, it would be best if you hire an attorney to assist you or, if you can not afford an attorney, contact your states' bar association for a referral and/or apply with legal aid.
Once the matter is properly filed and served, it will then be up to the court to determine whether allowing you visitation is in the best interest of the child
. Based on the information you have provided (the father, your son, is incarcerated, and you have a close relationship to the child), you may have a good case for visitation. Hope this helped, if you have a follow-up do ask.