Drupal - How do I avoid checking for updates for a specific module?

You need to implement hook_update_projects_alter().

Alter the list of projects before fetching data and comparing versions.

Most modules will never need to implement this hook. It is for advanced interaction with the update status module: mere mortals need not apply. The primary use-case for this hook is to add projects to the list, for example, to provide update status data on disabled modules and themes. A contributed module might want to hide projects from the list, for example, if there is a site-specific module that doesn't have any official releases, that module could remove itself from this list to avoid "No available releases found" warnings on the available updates report. In rare cases, a module might want to alter the data associated with a project already in the list.

The project key from the .info file is added by the packaging script on drupal.org to identify what project the module is from. The primary use is for the Update status module to monitor versions of installed packages and notify administrators when new versions are available.

You just remove or comment this line in .info file and Drupal stop checking for updates of this module.

Just providing a code sample to help the chosen answer:

function MYMODULE_update_projects_alter(&$projects){
    unset($projects['slug_of_the_module_you_want_to_disable']);
    //dsm($projects);  // view a list of projects
}

Enter the module name in the $projects[] variable to disable. If you're not sure what the slug is (will be lowercase and underscored), use that dsm() call or print_r() to print out a list of the modules.