It looks like you are not using the same variables in the SQL that you used the quote() method on. I really advise you to use placeholders instead. Something like:

my $sqlCmd = $dbh->prepare("
    UPDATE $table SET $column = ? WHERE $column = ?
");
for ($count = 0; $count < $EasySetDataCount; $count++) {
    $sqlCmd->execute(
       $NewEasySetDataNames[$count], 
       $EasySetDataNames[$count],
    );
}
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The placeholders mean you don't need to worry about quoting at all, you just say where the value should go (with the question marks in the SQL) and what data should be placed in those places (the items in the call to execute).