Imessages not syncing from iphone to macbook air (Big Sur)

Hello ceabea,

Thank you for reaching out to Apple Support Communities.

Let's sign out of iMessage and Sign back in using these steps:

"Set up iCloud for Messages on your Mac

1. In the Messages app on your Mac, do one of the following:

• macOS Ventura: Choose Messages > Settings, click iMessage, then select Enable Messages in iCloud.
• macOS 12 or earlier: Choose Messages > Preferences, click iMessage, then select Enable Messages in iCloud.

2.Select which email addresses and phone numbers you want to send and receive messages from on your Mac.Any messages currently stored in iCloud are now accessible on your Mac, and any new messages you receive on your Mac will be stored in iCloud.Note: If you want to see SMS and MMS messages on your Mac, see the Apple Support article How to forward SMS/MMS text messages from your iPhone to your iPad, iPod touch or Mac.

3.To see Messages stored in iCloud, open the Messages app.

Note: If you want to turn off Messages in iCloud on your Mac, deselect Enable Messages in iCloud. You can choose to disable Messages in iCloud for all your devices, or just your Mac.

For more information on how to use Messages on your Mac, see the Send a message on Mac."

More details can be found here: Set up iCloud for Messages on all your devices

If the issue persists, can you provide a few more details? When did you first notice this? Also, what is your exact macOS version?

Cheers!