if user has posted in a specific forum

[PinkMilk]

Theory code only, not tested...

-- 1 --
Assuming single forum per client (rather than 1 forum multiple client threads) its not a navbar notification but is a possible alternative.

Change new posts link color

Give code a class rather than inline styling, throw in a little animation to cause it to flash.

.getmyattention {
  color:red;
  animation: blink 1s linear infinite;
}

@keyframes blink {  
  50% { opacity: 0; }
}

-- 2 --
Came up with this before above not really what your after but going to leave it here anyway rather then just delete.

Adds a getnew link to navbar to specific forum based on user id (does not check for, flash or give notification which is what your after):

Build a plugin using an Associative Array (userid => forumid)

PHP Code:

// fetch user id
$user_id = $vbulletin->userinfo['userid']; 

// user id => forum id
$forum_id = array(1=>"35", 2=>"37", 3=>"43");

// check exists and create link variable for navbar 
if (array_key_exists($user_id,$forum_id)) {

   $client_link = '<a href ="search.php?do=getnew&amp;forumid='.$forum_id[$user_id].'">Client Area</a>';
// add link to navbar
$vbulletin->templatecache['navbar'] = str_replace($template_hook['navbar_buttons_left'], $template_hook['navbar_buttons_left'].$client_link, $vbulletin->templatecache['navbar']);

  } else {

   $client_link =  '';

} 


then with each new client you just add to the array , 4=>"52" (userid => forumid)