So running this command gives the desire result

You don't say what the desired result is. Why not just use qq// until you have solved your problem whatever it might be. eg:

use strict;
use warnings;

use Test::More tests => 1;

my $args = qq#' -c "normal \r"'#;
my $cmd  = '/usr/local/bin/vim';
my $want = qq#/usr/local/bin/vim . ' -c "normal \r"'#;

is ("$cmd . $args", $want);
diag ("Tried for >$want<");
[download]

Modify as necessary until you can construct the actual string you want. qx// uses the same interpolation as qq// so use the latter when testing.

PS. Why are you using backticks when you are discarding the returned value anyway?