Number in Number-squared
Perl 6, 33 31 bytes

-2 bytes thanks to nwellnhof



{(grep {$^a²~~/$a/},1..*)[$_]}


Try it online!

Explanation:

{                            }  # Anonymous code block that returns
 (                      )[$_]   # The nth index of
  grep {          },1..*        # Filtering from the natural numbers
        $^a²                    # If the square of the number
            ~~/$a/              # Contains the number