Use Regexp::Common just for the case, that begin and end markers are nested.

#!/usr/bin/perl
use Regexp::Common 'balanced';
my $text = 'An abitrary BEGIN_MARKER string which END_MARKER could con
+tain BEGIN_MARKER just about END_MARKER anything' ; 
my $q;
$q = qr!$RE{balanced}{-begin => "BEGIN_MARKER"}{-end => "END_MARKER"}{
+-keep}!;

$text =~ s:$q:local $_ = $1; s/a/X/; $_:eg;
print $text;
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