First off, Montana was 1972, so there were only three. There may well be a connection between the three that I'm missing, but it also could easily have been a coincidence. States change constitutions all the time (Georgia has had eleven!). July 1 is a common date for things enacted by elections to go into effect, and 1970 was a midterm election year.

Illinois has had four constitutions, and the 1970 constitution was adopted by a special election (in 1970, but went into effect in 1971).

North Carolina's 1971 constitution was largely just a reorganization of their previous constitution, and it came about after a 1967 study by the NC Bar. NC has had three constitutions over the years.

Virginia has had six constitutions, and the movement for a new one there began in 1965. Wikipedia mentions this came partially as a reaction to the Civil Rights act of 1964 and the Voting Rights act of 1965.

All of the above information comes from state's constitutional wikipedia pages, which I hope is acceptable as I'm not positing any historical theory, just pointing out basic facts.