Good afternoon neubigb,
Welcome to the Apple Support Communities!
Congratulations on the purchase of your iPhone 11. It sounds like you are having an issue with Messages indexing on your device. How did you set up your iPhone 11? Did you transfer from another iPhone, restore from a backup, or set up as new? Are messages the only data that has not populated to your new iPhone? Do you still have your previous iPhone?
Try these initial troubleshooting steps:
If you can, quit the Messages app. Then turn the iPhone off.
Restart the iPhone and go into Settings > General > About and change the device name to "[Your name's] iPhone 11" to make it easier to differentiate between backups from your previous iPhone and this device.
Check to see how much storage you have available on your iPhone and in iCloud.
You may need to increase the amount of iCloud storage to accommodate your data.
About storage on your device and in iCloud
Update your iPhone to iOS 13.6 and restart to refresh it - Update your iPhone, iPad, or iPod touch. Then:
Update your carrier settings on your iPhone or iPad
Consider enabling Messages in iCloud to keep all of your Messages in sync.
If you still have your previous iPhone, you would enable it there first. If not, do you have a Mac or iPad showing all of your Messages?
Wait until the messages have uploaded to iCloud before enabling the feature on your new iPhone 11.
Keep all your messages in iCloud
Please reply with the additional information requested about your issue and we'll do our best to help!
Take care.
