How can I transfer an iMessage thread to my computer?

If you need to see a specific message conversation on your computer, you can take screenshots of the conversation and transfer them to your PC or Mac, either via email, or if you use iCloud Photos, you can see your photos, and download them here: icloud.com

You can see how to take screenshots here:

Take a screenshot on your iPhone - Apple Support

You cannot transfer an iMessage to your computer. If you use a Mac, you can send and receive your messages on it, and synchronize them with Messages in iCloud.

You can find more information about this here:

Use Messages on your Mac - Apple Support