Hello kal_girl, and welcome to the Apple Support Community.
Unfortunately, Apple doesn’t provide a built-in way to export an entire Messages conversation to a PDF, nor is there a way to download messages directly from Messages in iCloud through iCloud.com.
If Messages in iCloud is enabled, your complete conversation history should automatically sync to another Apple device signed in with the same Apple Account. From there, you can print or save the conversation as a PDF, but Apple doesn’t offer a one-click export feature.
Helpful links:
Change which apps sync and store data with iCloud - Apple Support
Set up iCloud for Messages on all your devices - Apple Support
Could you let us know whether you’re trying to export the conversation from an iPhone or a Mac?
Also, are you looking to export it for personal reference, or for legal or record-keeping purposes?
That may determine the most appropriate approach.
Please let us know the outcome.
Your feedback may also help other members of the Apple Support Community with the same question.
🙏🏻
Have a great day.
And one more thing…
Never forget to think different.