I have a Comodo SSL certificate on my host plan, however when accesing my site from google, it sends me automatically to
http://example.com, Where the green lock doesn't appear.
If I manually add "https", like: https://example.com it does show up!
Is there a way to access my website always with the green lock showing up? instead of manually having to write it everytime?
You can easily redirect to the https version of any page using rewrite rules/rewrite module of your web server (the exact way to do this depends on the webserver used). Ask your provider, this is a common case so there may even be a UI option in your console to do this.
Regarding google see this: http://webmasters.stackexchange.com/questions/67212/how-to-convince-google-to-list-https-version-of-website
It may also be good form to verify the protocol used to access the site in your authentication module and refuse authentication if the wrong protocol is used. Assuming web rules are used to redirect traffic this would to prevent leaking information due to a misconfiguration/bug.