If Jesus was born at a certain date in history, do Christians believe that the Trinity did not exist before that date? Alternatively, is some sort of incorporeal Jesus believed to have been in existence despite not yet having been born?
Christians believe that the Trinity consists of three distinct persons with one divine nature.
This Trinitarian Godhead exists infinitely and eternally, and therefore existed before time.
Christians also adhere to the very distinct doctrine of the Hypostatic Union, which is the belief that Jesus Christ has two natures.
One nature is truly divine. One nature is truly human.
These two doctrines are clearly stated in the first chapter of the Gospel of St. John:
The "Son" (2nd Person of the Trinity) existed "before all ages." The Son was born into the world of a virgin roughly 2000 years ago. His name is Jesus Christ.
The Nicene Creed puts it best:
There are, of course, exceptions, but most of mainstream Christianity believes that Jesus was eternal, that He always existed, coequal with God. The Trinity is God, and is Eternal.
Addressing Jesus' pre-existence:
Scripturally, the strongest support for this belief is John 1.
For some documentation showing that various mainstream denominations believe this...
A Google Search of "Jesus before incarnation" will turn up thousands of similar statements from myriad Churches and denominations. Of course, it would be inaccurate to say that all Christians believe this, it is a demonstrably widely held belief.
The details of what His pre-existence is like is unknown. However, most also believe that He made appearances in the Old Testament long before His physical birth. (This is known as a Christophany)
The LDS Church gives an entirely different view, still holding that He existed, but in the "Premortal life" that they believe.
As far as I know, God does not exists in time. Trinity always was and always will be (from human point of view). Trinity simply is (from God's point of view.. whatever that is) and Jesus just came to the world at specific point of time.
Just to add to the great answers by both David Stratton and Charles Alsobrook, That Jesus was both fully man and fully God is central to Christianity. Unless Jesus was a sinless man He would not be an acceptable sacrifice for our sins. That could only be possible if he were also deity, which by definition would be an eternal being.