The due dates for estimated tax payments are:
April 15
June 15
September 15
January 15
As mentioned, we often suggest that you consider making the 4th state estimated payment by December 31 so to get the deduction for that payment in the current tax year. This does not benefit people who pay Alternative Minimum Tax (as they get no benefit for state income tax payments).
The due dates relate to income earned through the previous month end. Accordingly the June 15 due date is to have cumulative payments relative to the cumulative income through May 31.
For example, if a person has a large gain on September 10, that gain generally does not need to be factored into the September 15 estimated payment but does need to be factored into the January 15 estimated payment.