I am somewhat sure that if a/b is expressed in lowest form then for a/b to be the square of a rational number, then both a and b must be squares.

The smallest answer I found was:

a = 1681

b = 144

a/b = 1681/144 = (41/12)^2

a/b + 5 = 2401/144 = (49/12)^2

a/b - 5 = 961/144 = (31/12)^2

The full answer is

a = 1681 * i^2

b = 144 * i^2

  where i is in {1,2,3,...}

(a ,  b)

(6724,576)

(15129,1296)

(26896,2304)

(42025,3600)

(60516,5184)

(82369,7056)

(107584,9216)

(136161,11664)

(168100,14400)

(203401,17424)

(242064,20736)

----------

def lowestForm(a,b):

    from math import gcd

    return [int(a/gcd(a,b)), int(b/gcd(a,b))]

big = 500

squares = [i**2 for i in range(1,big)]

print('(a ,  b)')

for a in squares:

    for b in squares:

        numer = lowestForm(a,b)[0]

        denom = lowestForm(a,b)[1]

        if numer + 5*denom not in squares:

            continue

        if numer - 5*denom not in squares:

            continue

        print('({},{})'.format(a,b))