You would have to install the root certificate that this service is using.

You can check the certificate chain as other browsers see this and export the root certificate and possible intermediate certificates and import these in Firefox. For the root certificate you will have the set the trust bit to use the certificate for websites.

Thanks a lot for your feedback. Problem solved.

I was able to export the "Microsoft Family Safety" root certificate from Chrome and import into Firefox Certificates Authority. This was issued on 11 June 2014 when I had upgraded the Windows 8.1 and it expires on 16th Jan 3011.

Once I imported the certificate I had to set the trust checkboxes for this certificate to issue further certificates for websites.

I am now able to get past the untrusted error when I try to go to https sites. Of course this was only happening for Firefox browser and only on the user accounts that were monitored by Microsoft Family Safety in Windows 8.1.

I also saw alternate procedure listed as known Firefox issue on Microsoft Support Known issues for Firefox users in Windows 8.1 Family Safety

   Log on to your child's account on each device where they use the Firefox web browser.
   Open the Firefox browser, go to https://www.google.com or https://www.yahoo.com. You should see warning page stating that "This Connection is Untrusted."
   Swipe in from the right edge of your screen to display the charms, and then click Search. Search for certificates.
   Click Manage user certificates.
   In the pop-up window, select Trusted Root Certificates Authorities, and then select Certificates.
   Select Microsoft Family Safety Certificate on the right-side pane, and then right-click to display the menu. Select All Tasks, and then select Export.
   Continue through the Certificate Export Wizard:
       When you are asked to export the Private Key, select No.
       For the export file format, select DER encoded binary X.509 (.CER).
       Use the browse button to browse to your desktop and specify a file name such as Family Safety Certificate for the file that you are exporting. The wizard automatically adds a ".cer" extension.
       Click Next, and then click Finish to complete the export.
   Open the Firefox web browser, and then click Options from the Firefox menu.
   In the pop-up window, click Advanced, and then select the Certificates tab.
   Click View Certificates to display Certificate Manager, and then click Import.
   Select the "Family Safety Certificate" file that you previously exported, and then click Open.
   In the Downloading Certificate dialog box, select the Trust this CA to identify web sites check box, and then click OK. Click OK again to exit the Certificate Manager.

Thanks for the suggestions.