I see double!

Since they are twins they will have same ages . Thus the only square no. between 30 and 40 is 36 .

Let the product be x The key is that they are twins. Because of this, they have the same age. So, x is a square. The only square between 30 and 40 is 36. Therefore, the product is 36.

Ayon and Pranon must have the same age since they were born at the same time. The rest is just iteration and testing:

Python:

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age = 0
product = age * age
while product < 30:
    age += 1
    product = age * age
    if product > 30 and product < 40:
        print "Answer:", product
        break

Ayon and Pranon are 6 years old when the product is between 30 and 40, making the product $\boxed{6 \times 6 = 36}$ .