If he were telling the truth, there'd be a month beginning with a vowel with another such month 7 months away, but there is no such separation. The closest that comes is the 6 months from April to October or vice versa.
So the knave is lying in both instances, so there's a 7-month interval from a truth-telling month to another such month, as the lying nature of his current statements make the months referred to as truth-telling months. Furthermore, the month at the beginning of that 7-month span must be followed by a lying month.
There are only 3 lying months, so let's see:
April is preceded by March, and 7 months after that is October, so that span is no good.
August is preceded by July, and 7 months after that is February. This works.
October is preceded by September, and 7 months after that is April. So, again, that doesn't work.
The only month that works is August and the knave is lying.
From The Mensa 365 Brain Puzzlers 2011 Calendar, from Page-a-Day publishing, by Mark Danna and Fraser Simpson. Puzzle for January 23.