The Survey Quotas "Quick CSV report" feature is broken in the current version of Limesurvey.

It's due to a typo in the actionQuickCSVReport() function of the application/controllers/QuotasController.php file:

        $surveyid = sanitize_int($surveyid);
        if (!Permission::model()->hasSurveyPermission($surveyId, 'quotas')) {
            throw new CHttpException(403, gT("You do not have permission on this survey."));
        }

PHP variable names are case-sensitive, and $surveyId is a non-defined variable, so nobody ever has permission to use the feature.
I changed that to $surveyid (the variable as named everywhere else in the file) on my local copy and the feature works properly again.