This is the basic idea of what I'd do:

my $count = 0;
while (<IN>) {
  s/^"\d+"$/'"' . ++$count . '"'/e;
  print OUT;
}
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...or, from the command line

% perl -i.bak -pe 's/^"\d+"$/q(").++$count.q(")/e' myfile
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The one-liner puts the original file in myfile.bak.