Can I apply to international internship 3 month early? I am a 2nd year masters student here in Morocco. To get my diploma I have to enroll in a 6 months internship from January to the end of June.

I found in the website of a well known company which is active in the field of my master an internship opportunity titled "Intern, Software Engineering" but based in Austin, TX, U.S.

My questions are: should I apply now? Or should I wait until December?
Should I specify that I need a 6 month internship starting from January?

Remark: The offer page contains only two buttons, "Apply with Linkedin" and "Apply Now". I didn't find any contact information to get more info.
I think you need to apply early as there is some delay in getting things set up properly before the internship begins. However, the question about whether you can start the internship early can only be answered by your current institution, which has made the rule. If they say it can't be started early, then the question becomes whether the employer wanting an intern is willing to wait for you to start and whether it will be sufficiently long for your needs. 

But you need to start asking those questions at the appropriate place (not here) immediately. Don't wait until December to start the process.