Actually, "let's see what's trending on mastodon." doesn't sound wrong to my ears at all, but maybe that's because I've been around people who say that for a couple of months.
For reference, the community-decided verb/noun for posting on Mastodon is "toot" (as in, the elephant sound), so that would be "can you toot me". I mean. It's ridiculous, but I like it. It's endearing.
In America (and I may be biased b/c my kids say it all the time) to toot means to blow your car horn...but it also means to fart. If I say "I tooted this morning," most people will assume I farted. You might want to reconsider that one.
Mastodon is the software. Welcome to the world of community powered social media. You could name your site twatter if you want, or quitter ;) and still use the mastodon software.
"Can you mastodon me?" Ugh.
I don't see this going mainstream with a name like this. I don't think anyone at NBC is ever going to say, "let's see what's trending on mastodon."