No, it won't. Quick start doesn't work very well right now and basically just transfers over your photos and does a clean install of your apps on the new phone.
I just transferred my iPhone X to a new iPhone 13 Pro and lost all my Authenticator 2FA keys on Google Authenticator. They are still on the old phone but my new phone shows an empty authenticator screen. My advice is to avoid using Quick Start and instead use the backup and restore feature. The backup and restore feature is at least known to work properly.