Update (February 27, 2019):

Google have announced domain-wide data in Search Console, which combines all subdomains and protocols of a domain, http, https, www, and non-www.

Old answer:

Google Webmaster Tools treats HTTP and HTTPS as separate sites and you cannot inform Google using Google Search Console, that you moved from HTTP to HTTPS since the address move tool does not support protocol changes

You can though add the HTTPS site versions. When adding a site, simply type the URL including the https:// prefix. You will then have two sites in Google Webmaster Tools.

This fits with the general principle that http://www.example.com is considered a different URL from https://www.example.com and may validly serve different content, Google is OK with that. In practise, most websites set up a redirect, but Google Webmaster Tools is designed to work in either case, just add both sites.

Similarly, you should add the www and non-www versions of your site, so in the end, you should have four sites:

• http://www.example.com
• https://www.example.com
• http://example.com
• https://example.com

You can set whether www or non-www is preferred, but I'm not aware of a similar setting for HTTP vs HTTPS.

The address move tool does not support protocol changes

Since HTTPS and HTTP are protocols you are not moving your site from one address to another, you are merely changing the URL path. Google Webmaster Tools does not support address moves on changes to the URL paths including sub domains, and protocols.

SOURCE

3. Request an address change.

   Use the Change of address tool when your site move entails a domain or subdomain change, such as changing from http://fish.example-petstore.com to http://example.com or http://example-petstore.com.
   Note: The tool does not currently support the following kinds of site moves: subdomain name changes, protocol changes (from HTTP to HTTPS), or path-only changes.

Therefore you do not need to inform Google Webmaster Tools that you are changing protocols, however it is recommended that you setup redirects from all old URLS to the new URLS, and update your sitemap to include the new URLS.

Add all variations of your site to WMT

While the site address move tool may not treat protocols, url changes and sub domains as new sites, the rest of Webmaster Tools does treat protocols and sub domains as separate sites. You should add all variations of your site, below is an example of my site BYBE added to WMT with all variations, you should do the same. (recommended by John Mueller from Google, See comments below this answer).

301 redirects recommended by Google

If you plan to serve the website as partial ssl or complete then you should setup good redirects, as recommended by Google:

SOURCE

Prepare for 301 redirects Once you have a mapping and your new site is ready, the next step is to set up HTTP 301 redirects on your server from the old URLs to the new URLs as you indicated in your mapping. Keep in mind the following:

• Use HTTP 301 redirects. Although Googlebot supports several kinds of redirects, we recommend you use HTTP 301 redirects if possible.
• Avoid chaining redirects. While Googlebot and browsers can follow a "chain" of multiple redirects (e.g., Page 1 > Page 2 > Page 3), we advise redirecting to the final destination. If this is not possible, keep the number of redirects in the chain low, ideally no more than 3 and fewer than 5. Chaining redirects adds latency for users, and not all browsers support long redirect chains.


• Test the redirects. You can use Fetch as Google for testing individual URLs or command line tools or scripts to test large numbers or URLs.

Setting up the redirect in Apache

Setting up redirects in Apache, ngInx, IIS is pretty straight forward, below is examples of redirecting 301 from HTTP to HTTPS in Apache2 .htaccess file.

SOURCE

Enforce SSL on specific pages and disable on rest

This script will remove SSL on all other pages part from the login page and register page, you can add more just use | as the separator between file names.

mod_rewrite:

RewriteCond %{HTTPS} on
RewriteCond %{SCRIPT_FILENAME} !\/(login|register)\.php [NC]
RewriteRule ^(.*)$ http://%{HTTP_HOST}/$1 [R=301,L]

Enforce SSL on the entire site

If you want to enforce SSL on the complete site then you can use mod_rewrite to detect HTTPS off.

mod_rewrite:

RewriteEngine On
RewriteCond %{HTTPS} off
RewriteRule (.*) https://%{HTTP_HOST}%{REQUEST_URI}

Sitemap changes

Since you are changing protocol you need to add a new property to Google as HTTPs, this will have no sitemap submitted as default, you will need to ensure that your sitemap contains all the new URLS and then submit it under the HTTPS property variation.

If you are moving your site from HTTP to HTTPS, you will need to submit the HTTPS site as a new site (defined as new property). I was advised that I did not need to do this, however Google WMT stopped reporting index status, crawl errors and traffic.

Once I added the HTTPS to Google Webmaster Tools I started receiving correct data within the Google's WMT console.

Google change of address tools: Note: The tool does not currently support the following kinds of site moves: subdomain name changes, protocol changes (from HTTP to HTTPS)

Google has a site migration howto. A protocol change counts as url modification; this is the relevant section.

The switch goes much like a regular HTTPS switch. Once your urls are made protocol-relative, and before you start redirecting, you should validate the https site in Google's webmaster tools. It's a separate site with its own sitemap. There is no need for other Google-specific steps, but you can use their tools to validate that traffic is carrying over fine.