Quick start won’t pop up

It may not if you've already entered your Apple ID and Password.

Do this now that the phone is updated and start over. You're going to wipe the phone back to the hello page as though it was fresh out of the box.

See the directions in this link --> What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch - Apple Support

Once you are at the Hello Page, do as the phone asks you to do. After you've signed into Wifi, etc., it will ask if you want to set up your phone from a phone you have nearby. Choose that option.

And there is the issue. The new iPhone MUST be update to 14.4.2. You can't transfer the date of a phone with a higher iOS version to a phone with a lower iOS version. Update the new phone, then use Quick Start.

OK....what is the new iPhone model and iOS version and what is the old iPhone and iOS version?

Did you follow the directions in the Related Article which is at the top of this page?