Jesus and John WERE CONTEMPORARIES of approximately the same age, so why would Herod believe such a thing?
For follow-up discussion and general commentary on the topic. Comments are sorted chronologically.
Herod Antipas reigned over Galilee and Perea from 4 B.C. to A.D. 39 (Lk. 3:1-2). The infamous woman "Herodias" was both niece and wife to Philip and Herod, being the daughter of Aristobulus, son of Herod the Great. She first married Philip, her uncle, by whom she had Salome. Later she left him to live publicly with her brother-in-law, who had been married before to the daughter of Aretas, king of Arabia Petra. John warned of him that it is NOT lawful because she was his niece and his brother's wife (Lev. 18:6,16; 20:21).
Either on the day of his birth or the day he began to reign, both were counted as birthdays by kings (Hos. 7:5; 1Sam. 13:1), the daughter of Herodias danced before them, and pleased Herod.
Whereupon he promised with an oath to give her whatsoever she would ask.
As she was instructed of her mother, she asked, "Give me here John Baptist's head in a charger". Persian kings always granted requests that were presented to them during such entertainment (Est. 5:3; 7:3).
What a wickedness! This incestuous, bloody, adulterous, depraved mother! She was determined to rid herself and her husband of this troubler of conscience. Lived in misery because of his fear of the people and because of the torment of his conscience by the preaching of John. In the end he feared the people more than God, so killed John (Mt. 14:1)
John spent about 18 months in prison before he was murdered.
Jerome says Herodias was so wicked that, after gloating over the head of John, she drew out his tongue and pierced it with a needle. Aretas, her father made war on him and destroyed Herod's army, which Josephus says was judgment on him for murdering John the Baptist (Antiquities, 18.5.2).
In fact, Herod had heard and obeyed many things John the Baptist told him (Mk. 6:20). Herod's troubled conscience of murdering of John the Baptist (Mt. 6:14; Lk. 9:7-9) made him to think that Jesus as John the Baptist; he is risen from the dead; and therefore mighty works do shew forth themselves in him. Mt 14:2
Although the Bible doesn't say why, John the Baptist and Jesus were actually cousins and therefore they may have looked quite similar to each other.