When I click "Realms" on the main menu, I get sent to an error message that reads, "Could not connect to Realms:java.security.cert.CertificateException: No subject alternative DNS name matching pc.realms.minecraft.net found.

Are you using the Java Runtime provided by the launcher or are you using a custom one (if so which)?

I do not believe I am using a custom Java Runtime. How would I be able to check? 

You can check it like this:

1. Opening the "Installations" tab in the launcher

2. Hovering over the installation you are using

3. Clicking on the three dots on the right and then "Edit"

4. Click on "More options"
   If "Java executable" is blank ("<Use bundled Java runtime>") then you are not using a custom one

Ok, I see that. It only says "Use bundled Java runtime," 

Could you please

1. Open pc.realms.minecraft.net in a browser

2. Follow these steps to export the certificate: https://www.shellhacks.com/get-ssl-certificate-from-server-site-url-export-download/

3. Attach the exported certificate

Do you have Firefox installed? Otherwise it seems there is not an easy way (without using the command line or scripts) to get the server certificate.

Where is the link supposed to take me? The official Minecraft website?

No the link is the one reported on the error screen shown in the screenshot you attached. I assumed that for some reason the server your Minecraft client is communicating with is not sending valid certificates, but since you cannot easily download the certificate with all browsers for this website I cannot verify this theory. And even if that proved correct we could likely not help you with that here on the bug tracker, thanks anyways for your cooperation.
I will therefore close this report as invalid.

Please contact the community support, linked below.

Quick Links:
📓 Issue Guidelines – 💬 Community Support – 📧 Customer Support – 📖 Game Wiki

Alright. Thank you for trying to help me. I appreciated it. 👍

Just an update for anyone who is concerned. It seems that my Minecraft realms functionality has been restored, although I'm not entirely sure how or why. The only thing I have done in the past week that was Minecraft related was just playing a single player world for about thirty minutes. Perhaps this error was remedied by playing single-player, an update that was personally experienced on my PC, or it is just dumb luck that It was fixed through the passing of many months.