Unless the words array is also needed for something else, my preferred solution would be to skip it entirely and just use a regex on the whole file content (what can I say, I like regexes):

use File::Slurp;
my $text = read_file('input.txt');

while ($text =~ /" (.*?) "/sg) {
  print "Found quoted string with ".split(' ', $1)." words: $1\n";
}
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The regex might need to be adjusted depending on the exact definition of what should be counted as a quoted string within the input data.