Applying for an internal role, do I tell my manager right away?

I just wanted to do a quick check with you guys on what the proper procedure here would be.

A while back (maybe a year or so) I worked with someone on a different team on a project and expressed to them an interest in getting into their role. They were great and took this on board and helped turn that project into a good learning experience for me. My manager said shortly after that this person had recommended me for that role, but there wasn't an opportunity to move at that time. They gave me the usual "let's see down the line" kind of response that I've noticed is rife here (sometimes to my benefit) and I came to accept it would never be revisited.

I happened to look at internal job vacancies this morning (something I hardly ever do) and noticed the perfect version of that job was posted late last month, I've decided I'm going to apply for it with the ideal situation being that I get it, but at the very least I'm interested to see what they say to me regarding whether I can move and why.

The job role is in a team belonging to my manager's manager, my plan was to just apply via the portal and call it a day until I hear back. Is that weird/wrong? Am I supposed to mention it to my manager now? I'm sort of just throwing it out there, not trying to plot corporate espionage, but If that's a big faux Pas it'd be good to know.

Just wondering whether it matters what I do here and what you'd do.

Thanks