If your name is not on the birth certificate, have you established paternity in some other way?  Do you have legal documents showing you are the father?  You will find information on the website for the Arizona Department of Health Services Vital Records section.  Go to  https://www.azdhs.gov/licensing/vital-records/index.php#birth-certificates-who-can-obtain .

If the child is more than a year of age, you will need a court order to change the child's name.  Those forms and instructions are available on the court's website.  Go to: https://www.azcourts.gov/selfservicecenter/Forms/Name-Change.  You may also want to talk to a lawyer if you do not have paternity documents.  There are three programs in Arizona that you can use to get inexpensive legal advice and guidance.  One is the modest means program.  You will need to apply for the program and meet the financial qualifications.  Once you are qualified, Modest Means will send you a list of attorneys whose practice covers your area of concern.  You select a lawyer from the list and will be entitled to a one hour consultation for $75.  If you need additional help and if the lawyer agrees to take your case, the most you will be charged is  $75 per  hour.  The telephone number is (866) 637-5341.  Both Maricopa and Pima Counties have lawyer referral programs that offer a 30 minute consult with an attorney.  Maricopa County's Lawyer Referral Service offers its 30 minute consult with an experienced attorney for $40.  The telephone number is (602) 257-4434.  Pima County's Lawyer Referral Service offers a 30 minute attorney consult for $35.  The telephone number is (520) 623-4625.  If you are unable to afford even a modest fee and you meet the financial need qualifications, you may be able to find legal help in another section of this website.  Go to https://www.azlawhelp.org/accesstojustice.cfm.