There are two answers: No and Yes.
The No answer: The expression “when the moon turns blue” was used for hundreds of years to mean “when hell freezes over” or “fat chance.”
“Blue moon” first appeared in print in the nineteenth century, but was probably used even before that because it’s a quirky idea and almost rhymes. There was no intention to connect either of these expressions with the moon’s actual behavior. (But people might once in a while have seen a real, blue-tinged moon caused by smoke in the air.)
The Yes answer: Whenever there are two full moons in the same month, the second one is often referred to as a blue moon. Calling it that is a very recent development. It dates from a March 1946 article in the astronomy magazine Sky and Telescope, based on an article in the Maine Farmer’s Almanac that had appeared ten years earlier.
The editors of Sky and Telescope have recently admitted, however, that they misinterpreted the Maine Farmer’s Almanac article and that the title “blue moon” was really meant to be bestowed upon the fourth full moon in any season. Seasons are three months long, so they would ordinarily have only three full moons.
That makes a big difference. The fourth full moon in a season is not necessarily the same full moon as the second one in a month; it might happen to fall in a month all by itself. But the fourth-one-in-a-season concept is not as easy for people to grasp as simply counting the number of full moons in a month (anybody can count up to two), so I predict that the second-full-moon-in-a-month “blue moon” will never die, no matter what the astronomers say.
It isn’t very unusual for two full moons to fall in the same month; it happens about four times a year, much more frequently than a fourth full moon in a season, which really does happen only once in a blue moon, every two and a half years or so.
Here’s how two full moons can occur in a single month. As you know, our calendar contains eleven 30- or 31-day months plus February. But the lunar month, the time it takes the moon to circle Earth (you know, of course, that it does) and return to the position in which it is totally illuminated, full-looking, is only about 29½ days.
So two of those 29½-day illuminations can easily fall within the same 30- or 31-day period. It can never happen in February, though, because at only 28 or 29 days, February is shorter than the lunar month.