the phrase "As of next month" means the next month after now, or the next month after the moment at which the speaker is speaking. So, if on may 15, someone says:
As of next month, all the airline's fares will be going up.
that means the fares will increase in June, probably at the start of June. However "As of the next month" means the next month after some specified point in time. So in May, a person could say:
They move into the new building in July, and as of the next month, all their prices will increase.
This means that the move happens in July, and the price increase in August.
Similarly, "next Monday" means the first Monday after the current day, but "the next Monday" means the Monday after a specified day, as in:
I start work Wednesday the 18th, and I have a meeting with the team on the next Monday.
(that would be Monday the 23rd.)