Given an array @words like shown in the question, you could do:

my $count;
foreach (@words) {
  my $quote = ($_ eq '"');
  if ($quote ... $quote) {
    if (!$quote) { $count++ }
    elsif ($count) { print "$count quoted words\n"; $count = 0 }
  }
}
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Although I admit that the ... operator is slighty obscure... :)