Can I view a list of the communities of a lemmy instance while logged into my own?

Thanks for asking this. Until I read your question, I didn’t realize it but I had been wondering the same

No it isn't. I opened a feature request for specifically this: https://github.com/LemmyNet/lemmy-ui/issues/1397

Yes, you could just go to https://startrek.website/communities for the list of communities on that particular instance.
However, subscribing to a community on an instance where you do not have an account takes a few extra steps.

When looking at the list of communities on another instance where you do not have an account, clicking the Subscribe link from the list will result in a popup stating that you're not logged in and if you click on the community name, a Subscribe button will not be shown.
However, most, if not all, communities will have instructions in the sidebar for how to subscribe from your main instance without the need for creating an account on the instance where that community resides.

To do this using the example of the startrek.website instance, if you wish to subscribe to the daystrominstitute community, you would enter "!daystrominstitute@startrek.website" into the search field on your main Lemmy instance where you are logged in.
Generally the first item in the results will be a link back to that community, which will look like:
"Daystrom Institute@startrek.website - 448 subscribers"
Click that link and you will see the content of that community again, but now notice that a Subscribe button will be shown in the sidebar.
Clicking Subscribe may result in a note that says "Subscribe pending", which I believe is due to some syncing that needs to occur between instances. Just give it a few minutes and then that community will appear in your list of subscribed communities.
You can now read, post and reply to that community just as if you had an account on that instance.

https://startrek.website/communities?listingType=Local&page=1

That link might work, they have 4 communities