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When you specify Https to connect to a website, the actual connection requested by the browser is different. That’s something that you need to set up on your site – and you’re quite correct. The HTML doesn’t need to change for that. Instead of using https on only sensitive parts of your site, you may go ahead and use https for your entire site. This does not affect accessibility or performance in any way and it’s a great way to hedge your bets against future Google algorithm changes. Changing the text “http” to “https” in all of your links that point to other parts of your own site is likely the one technical task you’ll need to do yourself. You should set Up 301 Redirects.

Nah moz is smart enough to auto detect the changes and adjust accordingly. Make sure all your 301 redirect are in place and over a week or 2 moz should begin listing the https as google index's then and replaces your http.