tl;dr– No, the claim about high IQ's appears to be based on an early definition of IQ that's no longer used. It's misleading to use this outdated definition of IQ as the claim would be very different (and far more impressive) if it were based on the current definition.
So, using Google Translate on their webpage (as it's in Japanese):
Intelligence Quotient (IQ) and Achievements
IQ (IQ) is
(Score of intelligence test) ÷ (average of score of same age of child) × 100
In the infant's IQ examination,
(Age calculated from the degree of child's development) ÷ (age of children) × 100
—"About intelligence and IQ", translated by Google Translate from Japanese to English
In other words, if a 1-year old scores as well as a 2-year old, then they'd have an IQ of 200.
This leads to their claimed results:
||Number of people
|More than 1 month - less than 6 months
|From 6 months to less than 12 months
|From 12 months to less than 24 months
|24 months or more
and then their corresponding guarantee:
IQ 140 at 3 years old (average 3 years 7 months) that we guarantee means that the child is developing about 5 years old children. In our achievements, the IQ of a student who has been in the baby park for more than two years exceeds 150.
They're basically claiming that they can get their young students a-year-or-two ahead of their peers over the course of two years of enrollment. We haven't independently verified that claim, but it seems pretty plausible.
Inconsistency with the modern, common definition of IQ.
The problem is that this isn't how we define IQ anymore. Instead, we define IQ as being performance on the IQ exam cast to a normal curve with a mean/median of 100 and a standard deviation of 15.
In this definition, an IQ of 152.6 implies performance at (152.6-100)/15 ~= 3.5 standard deviations above the median peer rather than performing like the median peer who's ~1.5 times older.
The desire for an IQ metric has been based on the historical observation that some people are smarter than others and the desire to quantify it. In the early days, IQ was defined like the Japanese school claims, which is why it was called the Intellectual Quotient.
The problem with this old-fashion definition is that it's too mutable; it doesn't capture basic intelligence so much as preparation for a particular exam. It's this mutability that the school exploits in its claim.