The key is to chomp your data to eliminate trailing newlines. You can also use join to concatenate the array elements with a specified separator. Here's how I might do it:

#!/usr/bin/perl

use strict;
use warnings;
use feature qw/say/;

chomp(my @fields = <DATA>);

my $table_fieldname_array = "('" . join("','", @fields) . "')";

say $table_fieldname_array;

__DATA__
FIELD1
FIELD2
FIELDN
[download]

This outputs:

$ perl 1117442.pl
('FIELD1','FIELD2','FIELDN')
$
[download]

Which I gather is what you want.