No. Take the square of any number above 100, and you get a result above 10,000.
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The only perfect squares from 1 to 31 are 1, 4, 9, 16, and 25.All of the other 26 are NOT perfect squares.2,3,5,6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,27.28,29,30,31
All integers that are not perfect squares.
-90 squared is rational - it is +8100. All perfect squares are not only rational but they are integers.
I'm not sure exactly what your question is, but the squares of 4 and 5 do have this property (and are the only perfect squares that do).