The correct answer is: “Yes… and, no!” On the one hand, the Son of God always existed (John 8:58; 17:5), for He was eternally with God (John 1:1-2) and through Him God created all things (1 Corinthians 8:6; Colossians 1:16-17; Hebrews 1:2). So, yes, the Son of God existed before Jesus was born. But, on the other hand, to be strictly correct, “Jesus” did not exist before He was born, because the name “Jesus” was not given to the Son of God until AFTER He had taken on a human nature and was born into the world (Matthew 1:25). This is why, to be completely precise, the best answer to the question is “Yes… and no!”
As an analogy, think about marriage. In many countries in the world, when a woman marries, she will change her name. For example, in England, if Mary Smith married John Jones, her name would change to Mary Jones. If we ask the question, “Did Mary Jones exist before she got married,” we could answer “Yes,” because the same person still existed before she got married; but, if we wanted to be very literal, we would answer “No,” because before Mary Smith got married, she was not called Mary Jones. In a similar way, the Son of God existed both before and after He was born; but it was only AFTER He was born that this Divine Person could properly be called by the name of “Jesus”.