I don't know the specifics of Julia's GC, but this seems a strange thing to say in 2017. Douglas Crosher's conservative generational collector for CMUCL (also used in SBCL AFAIK) supports C interoperation and is entirely adequate for handling the extra garbage that (admittedly) is generated when using functional collections. I don't recall exactly when he wrote that collector, but it must have been 20 years ago at least. It would be strange if Julia weren't using something at least as sophisticated.
I didn't say Julia programmers don't have to worry about types, my comment was only about type annotations, though they are obviously related. Perhaps a better way to phrase it is: by worrying a little bit about types, you can generally avoid the need for type annotations. Hopefully as the tooling around the language matures, we can reduce the amount of worrying required.
I think it is a little disingenuous to say that a Julia programmer does not have to worry about types. Type inference alleviates many burdens, but correct typing of arguments is essential (and hidden promotions or casting can kill performance). So while you can write correct programs easily, for efficient programs you end up worrying about this quite a lot.