I was able to find the answer on Slack! Thank you to everyone for the help.
It seems like I didn’t have the “Basic” trust level badge on the Discourse yet. I mostly read the Julia Discourse without logging in so I guess I didn’t have enough “reading time” yet on the forum. But now I do! Once I read a few more topics while logged in it gave me the “Basic” trust level badge.
I can now see the “Message” button when I click on my profile or other people’s profiles.
Thank you to @Valentin and @giordano on Slack! Thank you to @PetrKryslUCSD as well!
EDIT: Link to the badges: JuliaLang