When this problem was first in the queue, it was unclear whether the word 'incremented' was meant to convey (1) addition or (2) concatenation.  The example makes it clear that what is meant is the latter.

I found results for both meanings, program output:

Interpretation (1) Addition
1 3 4 2
1 8 9 3
8 1 9 3
8 8 16 4
27 9 36 6
216 9 225 15
64000 9 64009 253
97336 8 97344 312

Interpretation (2) Concatenation
1 6 16 4
8 1 81 9
18821096 1 188210961 13719

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solutions1 = []
solutions2 = []
cubes = [n**3 for n in range(1,10**4)]

print('Interpretation (1) Addition')
for c in cubes:
    for d in range(1,10):
        if issquare(c+d):
            print(c,d,c+d,int((c+d)**.5))
            solutions1.append([c,d,c+d,int((c+d)**.5)])

print()

print('Interpretation (2) Concatenation')
for c in cubes:
    for d in range(1,10):
        incr = int(str(c) + str(d))
        if issquare(incr):
            print(c,d,incr,int((incr)**.5))
            solutions2.append([c,d,incr,int((incr)**.5)])