2018 answer: webextensions (which use the Chrome extension API) can do this in Firefox 62

You'll need to make a WebExtension, which is also called a browser extension.

See accessing security information on MDN

You can also check out the docs for:

• getSecurityInfo
• SecurityInfo
• CertificateInfo.

You'll need Firefox 62.

Here's a working background.js

var log = console.log.bind(console)

log(`\n\nTLS browser extension loaded`)

// https://developer.chrome.com/extensions/match_patterns
var ALL_SITES = { urls: ['<all_urls>'] }

// Mozilla doesn't use tlsInfo in extraInfoSpec 
var extraInfoSpec = ['blocking']; 

// https://developer.mozilla.org/en-US/Add-ons/WebExtensions/API/webRequest/onHeadersReceived
browser.webRequest.onHeadersReceived.addListener(async function(details){
    log(`\n\nGot a request for ${details.url} with ID ${details.requestId}`)

    // Yeah this is a String, even though the content is a Number
    var requestId = details.requestId

    var securityInfo = await browser.webRequest.getSecurityInfo(requestId, {
        certificateChain: true,
        rawDER: false
    });

    log(`securityInfo: ${JSON.stringify(securityInfo, null, 2)}`)

}, ALL_SITES, extraInfoSpec) 

log('Added listener')

manifest.json:

{
    "manifest_version": 2,
    "name": "Test extension",
    "version": "1.0",
    "description": "Test extension.",
    "icons": {
        "48": "icons/border-48.png"
    },
    "background": {
        "scripts": ["background.js"]
    },
    "permissions": [
        "webRequest",
        "webRequestBlocking",
        "<all_urls>"
    ]
}

It also may be implemented in Chromium once this code is merged.