Pyth, 15 bytes

}Qmsjk^R2d*SQUh

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Test cases

A program that takes an integer 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
           SQ   # of the range from 1 to Q
             Uh # with the range from 0 to Q