Perform the suggestions mentioned in the following articles:

• "This Connection is Untrusted" error message appears - What to do
• Secure connection failed and Firefox did not connect

Check and tell if its working.

No it will not let me bypass the warning

Make sure that the values of the following Preferences are set to True on the about:config page

• browser.xul.error_pages.enabled
• browser.xul.error_pages.expert_bad_cert

• http://kb.mozillazine.org/about%3Aconfig

You can check who issued that certificate.

You can retrieve the certificate to check who issued the certificate. You should always do that to see if that helps to identify the problem (missing intermediate certificate or something else).

-> click the link at the bottom of the error page: "I Understand the Risks"

-> Let Firefox retrieve the certificate: "Add Exception" -> "Get Certificate".

• click the "View..." button to inspect the certificate and check who is the issuer.

-> Only leave the checkmark in the box at the bottom to "Permanently store this exception" if you trust that certificate.

• click "Confirm Security Exception" to enter the site if you still want to go to that website.

Check and tell if its working.

"browser.xul.error_pages.expert_bad_cert" was set on false so I set it to true and tried reloading. Got the page giving me the option to add exception and continue. Kept trying that and would not continue to the page even after accepting 10+ times.

-> Clear Cookies & Cache

• https://support.mozilla.com/en-US/kb/Template:clearCookiesCache

-> Clear the Network Cache

Check and tell if its working.

Did both, no luck. Still getting the page with this (see below). I don't understand why it's letting me use FF to log in to Mozilla to reply to these...but I can't log into facebook, gmail, etc.

This Connection is Untrusted

         You have asked Firefox to connect securely to accounts.google.com, but we can't confirm that your connection is secure. Normally, when you try to connect securely, sites will present trusted identification to prove that you are

going to the right place. However, this site's identity can't be verified.

         What Should I Do?
         
           If you usually connect to this site without problems, this error could mean that someone is trying to impersonate the site, and you shouldn't continue.
       
         Technical Details
         accounts.google.com uses an invalid security certificate.

The certificate is not trusted because no issuer chain was provided.

(Error code: sec_error_unknown_issuer)

         I Understand the Risks
         
           If you understand what's going on, you can tell Firefox to start trusting this site's identification. Even if you trust the site, this error could mean that someone is tampering with your connection.
           Don't add an exception unless you know there's a good reason why this site doesn't use trusted identification.

In your Firefox Connection settings, what kind of connection is selected ??

No proxy

Try changing it to Auto-Detect Proxy Settings for this Network or Use System Proxy Settings.

Check and tell if it works.

Tried both and did not work.

Who is the issuer of the certificates?

You can retrieve the certificate and check details like who issued certificates and expiration dates of certificates.

• Click the link at the bottom of the error page: "I Understand the Risks"

Let Firefox retrieve the certificate: "Add Exception" -> "Get Certificate".

• Click the "View..." button and inspect the certificate and check who is the issuer.

You can see more Details like intermediate certificates that are used in the Details pane.

-> Troubleshoot extensions, themes and hardware acceleration issues to solve common Firefox problems

Check and tell if it works.

Starting in Safe Mode, did not work.

Create a New Profile as a test to check if your current profile is causing the problems.

• Basic Troubleshooting - Make a New Profile

Now browse the website on which you were having problems earlier. Check and tell if it works.

Created a new profile. I was able to log into Gmail but still had some features disabled (would not show contacts online). Also still could not log onto Facebook

-> Fix login issues on websites that require a username and password

Check and tell if its working.

You haven't told us who issued the rejected certificate, so can you please do that?

The problem was that the certificate was issued by ESET. I unistalled ESET Internet Security and re-installed the new version with parental control disabled. Now firefox opens gmail sign in just fine. I had no trouble before with IE, just with firefox 13. Now the google certificate appears as issued by Thawte SGC CA and not by ESET.

I've had problems with Firefox 13 and gmail/google accounts. In Firefox 13 the SPDY is enabled by default, previous Firefox versions had it disabled.

Once I disabled SPDY in Firefox 13, all my gmail and google accounts have no problems or errors but with it enabled, google can't handle it for some reason.

To disable, about:config and change:

network.http.spdy.enabled - set to false.