Yes, you do need to support it. You can search on this forum for lots of info on that subject.

My island at my last house was 66" x 38", 13" of that was overhang. I did not have brakets or corbels supporting it, and was told I wouldn't need them. We never had a problem, despite lots of kids and heavy entertaining. Our granite as standard 1 1/4" stuff.

Standard recommended support for an overhang on 1.25" material is 8", with a corbel or support every 30" of unsupported overhang.

I was told I needed supports, and I was told I didn't need supports. I have 11.5 inch overhang. I opted to go without corbels/supports. I hate them. IF we do NOT stand or sit on it, I was told I'd be fine even by the guy who suggested the corbels. I hate kneeknockers!

You do not need "knee knockers" to have adequate support. Ask your fabricator to enquire with a company called "Chemical Concepts" for their "Counter Balance" support systems. (See link)

They offer support without being obtrusive.

Here is a link that might be useful: Counter Balance brackets

I hate knee knockers and corbels too but I wanted some support for a 12 inch overhang. I opted for steel L brackets that you can't see. I got them from the internet - website and their pics of installation like mine is attached. The guy I spoke with was very helpful as well - definitely knew his stuff.

Here is a link that might be useful: Steel L brackets