Long-period records from the National Hurricane Center show that the month of June averages about one named storm every two years. This number has remained remarkably consistent throughout the reconnaissance era (starting in 1944) and the satellite era (starting in 1966).
The graphic above shows the formation points of 82 tropical (and subtropical) storms that have formed in the month of June since 1851. Preferred regions this early in the season include the western Caribbean, Gulf of Mexico and just off the southeast coast of the U.S.
Some folks have wondered if the early tropical storm activity in May (Alberto and Beryl) could signal more early-season activity in June. A quick check of the historical records shows 21 storms have formed in May over 20 separate years (one year had two May storms). Of those 20 years with storms in May, exactly half of the years had a named storm form in the month of June. Therefore, the average of having a named storm in June about every other year seems to hold regardless of the activity in May.