Have a web application at URL: EnterpriseJazz.com
I recently registered a shorter domain name: ejazz.cc in order to use as a url shortener.
I configured godaddy to forward the shorter ejazz.cc to EnterpriseJazz.com and it works fine for http (ie: I just typed ejazz.cc in the browser and EnterpriseJazz.com loaded without any problems)
When I tried to use it in in production with with an https url, the browser gave me the "Connection is not private" error.
Is there a way to change the configuration on GoDaddy to solve this or am I going to have to do something fancy with my SSL certificate?
While it's not 100% clear from your question, the assumption here is that HTTPS works for the first URL, but not the second.
A certificate is tied to an identity. In the case of a HTTPS certificate, this identity is the DNS name of the server/service being accessed. This DNS name is stored in the Subject Alternative Name (SAN) extension of certificates.
Your current certificate is valid for enterprisejazz.com, but if you're now going to use an additional name - ejazz.cc - you will need a modified certificate that also covers this new domain. Fortunately, the SAN extension can hold multiple entries.
Commercial CAs don't modify certificates, so you will need to request a new one containing both names. While you're at it, it might be worth planning ahead and adding all DNS names your final service will use to save rework later.
You can in many hosting servers tie multiple certificates to a hosting IP / port. You could tie both certs to that end point and allow the server to select the appropriate certificate for the domain or forwarded domain.
There is also the possibility of doing forward rather than a redirect or vice versa to have all URLs end up at a single destination (with the appropriate cert)