Yes Jesus did have a birth certificate according to Mosaic Law. From Luke chapter 2:
22 Now when the days of her purification according to the law of Moses were completed, they brought Him to Jerusalem to present Him to the Lord 23 (as it is written in the law of the Lord, Every male who opens the womb shall be called holy to the Lord),[f] 24 and to offer a sacrifice according to what is said in the law of the Lord, A pair of turtledoves or two young pigeons.[g] (NKJV)