No, Judas was not a relative of Jesus. The Judas who betrayed our Lord was named Judas Iscariot. This name is explained in John 6:71, “Judas the son of Simon Iscariot.” (ESV) In the Bible, he was never identified as a family member of Jesus.
Judas was a very common and popular name among the New Testament Jewish population. A Jew named Judas Maccabee had successfully led a Jewish revolt against the Greeks in 167–160 BCE. Judas was a great hero and thus many parents named their children after him. It would be silly to conclude that all persons named Judas are the same. Consider some other New Testament people with that name.
There was a second apostle named Judas, “Judas the son of James”, Luke 6:14-16. After Jesus’ resurrection, Acts 1:13 includes this apostle with the other ten: “And when they had entered, they went up to the upper room, where they were staying, Peter and John and James and Andrew, Philip and Thomas, Bartholomew and Matthew, James the son of Alphaeus and Simon the Zealot and Judas the son of James.” This person named Judas was not a relative of Jesus either.
Finally, there is another third prominent “Judas” in the New Testament: “Jude, a servant of Jesus Christ and brother of James” (Jude 1:1). He wrote the Epistle of Jude, dated between 70-90 A.D. The James mentioned in this verse is historically considered to be Jesus’ half-brother. If this is accurate, then this Jude, the author, would also be Jesus’ half-brother. But he could not be Judas Iscariot because that man had died in 33 A.D.
Jesus did have half-brothers and half-sisters. They are identified in Matthew 13:55: “Then they scoffed, ‘He (Jesus) is just the carpenter’s son, and we know Mary, his mother, and his brothers–James, Joseph, Simon, and Judas.’” However, this Jude was not called one of the apostles and was not Judas the son of Simon Iscariot.