I have the feeling you are confusing something here. Google Analytics does indeed show referral traffic from various sources, i.e. referral, organic, social, ... you name it. And some referral traffic might be from sites/application that has TLS/SSL adopted (https). However, this has nothing to do with whether or not you are on https or HTTP.
You certainly should go to the Analytic settings of your Property and change the default URLs property to HTTPS instead of HTTP. And that's it.
When you are setting a 301 at the web server level (Apache, Nginx) from HTTP to https, it will be sufficient for Google to redirect all existing links to your new site. There are various discussions whether or not how much link juice is lost by switching to HTTPS and then forwarding - but generally speaking it is the right move.
There will be temporary loss in traffic (i.e. when HTTP URLs drop out of SERPs and until https replaces it - 3 to 5 days on average) and then the fact that rankings will temporarily be lost. Make sure you have a product/service people need, content that covers all searchers intent and reflects sales funnel, as well as an effective outreach campaign - and you will be just fine :)
Hope this helps!