I think this is referring to lunisolar (as I've usually seen it written) cycles where there are very nearly whole numbers of lunar months in a period of years. The best known of these is the Metonic cycle of 19 years, in which there are 235 lunar months (cycles of the moon's phases). For example, there will be a full moon on November 30, 2020, and also nineteen years from now on November 30, 2039.
There are lots of other periods of years where the lunar phases would repeat on about the same day, with allowances for leap years, etc. It looks like periods of 2300 years and 1260 years would come reasonably close to comprising an integer number of lunations - although not as close as the 19 year Metonic cycle (there are 60 periods of years shorter than 2300 years that are in closer sync than the 2300 year period).
Some internet searching showed that the numbers 2300 and 1260 are found in the Bible - 2300 in Daniel, in reference to the number of days that a sanctuary would be desecrated, and 1260 in Revelation, (the number of days for which some sackcloth-clothed witnesses prophesied, and also how long the dragon's mother was nourished after she fled to the place prepared by God for her in the wilderness). In the 18th Century, a French astronomer (and apparently, minute reader of the Bible) Jean-Philippe Loys de Cheseaux noticed that these two numbers were close-ish to a lunisolar cycle, and that (crucially?) their difference (1040) is also a close lunisolar cycle (about 15 minutes better than the Metonic cycle, but not so close as some shorter cycles).