Previous advice from others about using a module notwithstanding, this additional requirement puts it well into "If it gets any fancier" territory. Here, therefore, is the test:

use strict;
use warnings;

use Test::More;

my @good = (
    '"18/02/2018"',
    '"""18/02/2018"'
);

my @bad = (
    '""18/02/2018"'
);

my $re = qr/^"("")?[^"]/;

plan tests => @good + @bad;

for my $str (@good) {
    like ($str, $re, "$str matched");
}
for my $str (@bad) {
    unlike ($str, $re, "$str not matched");
}
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