This is quite cute...

use 5.010;
use strict;

QTRIM_SETUP: {
    my $trimmer;
    BEGIN {
        $trimmer = sub ($) {
            @_ = map { s/(^\s+)|(\s+$)//g; $_ } split /\n/, shift;
            shift until length $_[0];
            pop until length $_[-1];
            join "\n", @_;
        }
    }
    use PerlX::QuoteOperator qtrim  => { -emulate => 'q',  -with => $t
+rimmer };
    use PerlX::QuoteOperator qqtrim => { -emulate => 'qq', -with => $t
+rimmer };
}

my $string = qtrim{
    Hello
    World
};

say "[$string]";
[download]

Obviously, if you were doing this in "real life" you'd move the QTRIM_SETUP block into a module, so that you could just do:

use 5.010;
use strict;
use PerlX::Qtrim;

my $string = qtrim{
    Hello
    World
};

say "[$string]";
[download]

There's a slight annoyance with qqtrim in that this:

my $embedded   = "\n  \n";
my $multiline  = qqtrim{
    Hello
    $embedded
    World
};
[download]

... will also end up stripping the spaces in $embedded which is somewhat counter-intuitive.