Why are the flowers falling off the zucchini plant and not producing a zucchini?
This is typical behavior for a squash plant(and cucumbers). They will produce mostly male (non-fruiting) blossoms early in the season and will gradually start to produce more female blossoms as the season progresses. Give it some time and you will see it start fruiting.
If you are seeing fruit but they are falling off, this article will help: