There are several topics and posts discussing this in this forum.
This one here and the follow-up may cover what you seek?
You can use Julia running in JupyterLab right now in your browser via the repo here.
To get what you seek, do not simply use the launch binder badge. Using that will default to opening in the classic notebook interface. There is a way to edit the URL to switch to lab from there, described here.
However, it is easiest to just use the following link in your browser address bar, which is an altered version of what the launch badge actually uses, to launch JupyterLab directly:
https://mybinder.org/v2/gh/binder-examples/demo-julia/master?urlpath=lab/tree/demo.ipynb
If you paste that URL in your browser and open the page, JupyterLab will spin up with the Julia demo notebook already in the main panel on the right.
There’s also this binder example featuring Julia in Jupyter, but the kernel version is rather old, and so I didn’t test it further.