Pyth, 1415 bytes
}Qmsjk^R2d*SQUQmjk^R2d*SsQUs
A program that takes an integer value as a string (because the trailing 0 is recognized as a a separate value) and outputs True
for the 'concats of two squares', False
otherwise. The calculation time for the large values exceeds 1 minute.
Explanation:
}Q # check if the input value is in the list.
m # The list is constructed by mapping:
s # numeric conversion of the strings,
jk # concatenation
^R2 # and squaring
d # of the argument d
* # mapped over the cartesian product
SQSsQ # of the range from 1 to Q
U Us # with the range from 0 to Q-1