A quarter of a very lopsided orbit
Pluto takes about two hundred and forty eight years to circle the zodiac, so a quarter of the way round would nominally take sixty two years. The word nominally is doing heavy work there, because Pluto's orbit is by far the most eccentric of the bodies astrology uses.
At its closest to the Sun it is inside the orbit of Neptune and racing, clearing a sign in about twelve years. At the far end of its path it crawls, and a single sign can take more than thirty years. The quarter turn is a quarter of the circle, not a quarter of the time.
Pluto's demotion from planet in 2006 changed nothing for astrological practice, which had never depended on the classification. The orbit is the same and the ephemeris is the same.
Why the age varies more here than anywhere else
Because the speed changes so much, the age at which a person meets this square depends heavily on the sign and degree Pluto held at their birth. Charts with Pluto in the fast stretch, roughly Leo through Scorpio, reach the square in the late thirties and forties. Charts with Pluto in the slow stretch wait into their fifties or sixties.
For some birth years the answer is that it never arrives at all. That is a genuine result rather than an error, and the widget above will report it plainly when the transit is not reached within the years it searches.
People born in the middle of the twentieth century sit near the fast end of the orbit and generally meet the square young. Those born earlier, with Pluto in Cancer or Gemini, may find the angle sits beyond any plausible age, and the honest answer for them is that the transit does not occur.
Passes, and how long Pluto stays close
Pluto is retrograde for around five months of each year and moves only a degree or two forward in the rest of it. Three exact crossings across roughly a year is the usual pattern, and the planet often stays within a degree of the angle for a year or more on either side of that.
This makes the notion of a single date less useful than it is for a faster body. Astrologers read the whole approach as the transit, and each crossing is listed above so the shape of the span is visible.
This also means the number of crossings varies with where in the orbit Pluto is moving. Faster stretches produce a cleaner three pass pattern, while the slowest stretches can keep the planet in range for an extraordinarily long time.
What is read into it, and where
Pluto is conventionally associated with power, compulsion and things that are not reversible once begun. The square is described as the point at which whatever the natal Pluto stands for is confronted rather than managed. Because the age varies so widely, no single life stage attaches to it, which is a useful corrective to the neat midlife stories told about Uranus and Neptune. The tradition allows for the harsh readings as well as the constructive ones, and nothing on this page predicts an event.
The house of your natal Pluto is where the reading concentrates, and a birth time is needed for it. The dates do not require one.
Because of the spread in ages, comparing your dates against those of people born a decade either side is more informative here than for any other transit. The differences are large enough to see at a glance.
Questions people ask about this
At what age does Pluto square Pluto?
Anywhere from the late thirties to the sixties, depending on where Pluto sat when you were born. No other transit in common use has a spread of that size.
Why is the range so wide?
Pluto's orbit is highly eccentric, so its speed through the zodiac varies enormously. It can cross a sign in about twelve years at one end of the path and take more than thirty at the other.
What if I never get one?
That happens for some birth years and it is a real answer, not a failure of the calculation. The widget above says so directly when the angle is not reached within the span it searches.
How long does the transit last?
Pluto moves so slowly that the exact crossings often spread over a year and the planet stays near the angle for a good while either side. Treat it as a period rather than a date.
Do I need my birth time?
Only for the house placement. Pluto's degree does not change meaningfully across a day, so the dates are the same with or without a recorded time.
Is a Pluto square dangerous?
The tradition reads it as demanding and hard to reverse, not as an omen. This page states the dates and the house and marks the rest as interpretation, so nothing here is a forecast.