Hey dyconnor,
If you're using iCloud for Messages, you will not be able to see the individual messages in iCloud.com however they should sync to the new iPhone once you sign into iCloud in Settings > [your name]. After signing in, make sure Messages is enabled by going to iCloud > Apps Using iCloud(Show All) > Messages. For more details about iCloud for Messages, review this article: Keep your messages up to date with iCloud.
Since you do have the old iPhone, you have the option of turning on iCloud for Messages, allow them to sync to iCloud, which will then sync them to the new iPhone 15 Pro Max.
If you were not using iCloud for Messages, the messages will be included in an iCloud or computer backup. After restoring the backup, the messages that were on the iPhone at the time the backup was created will be restored to the new iPhone. Keep in mind, when restoring from a backup, all data currently on the iPhone will be replaced with the information that is in the backup.
If you do not have a current backup of your iPhone 14 Pro Max, you may do so by following one of the steps below or in this article: Back up iPhone.
Back up iPhone using iCloud
1. Go to Settings 
> [your name] > iCloud > iCloud Backup.
2. Turn on iCloud Backup.
iCloud automatically backs up your iPhone daily when iPhone is connected to power, locked, and connected to Wi-Fi.
Note: On models that support 5G, your carrier may give you the option to back up iPhone using your cellular network. Go to Settings > [your name] > iCloud > iCloud Backup, then turn on or off Back Up Over Cellular.
3. To perform a manual backup, tap Back Up Now.
To view your iCloud backups, go to Settings > [your name] > iCloud > Manage Account Storage > Backups. To delete a backup on the device you’re using, choose a backup from the list, then tap Turn Off and Delete from iCloud. To delete a backup on a different device using the same Apple ID, tap Delete.
Note: If you turn on an app or feature to use iCloud syncing (in Settings > [your name] > iCloud > Show All), its information is stored in iCloud. Because the information is automatically kept up to date on all your devices, it’s not included in your iCloud backup. (See the Apple Support article What does iCloud back up?)
Back up iPhone using your Mac
1. Connect iPhone and your computer with a cable.
2. In the Finder sidebar on your Mac, select your iPhone.
To use the Finder to back up iPhone, macOS 10.15 or later is required. With earlier versions of macOS, use iTunes to back up iPhone.
3. At the top of the Finder window, click General.
4. Select “Back up all of the data on your iPhone to this Mac.”
5. To encrypt your backup data and protect it with a password, select “Encrypt local backup.”
6. Click Back Up Now.
Note: You can also connect iPhone to your computer wirelessly if you set up syncing over Wi-Fi.
Back up iPhone using your Windows PC
1. Connect iPhone and your computer with a cable.
2. In the iTunes app on your PC, click the iPhone button near the top left of the iTunes window.
3. Click Summary.
4. Click Back Up Now (below Backups).
5. To encrypt your backups, select “Encrypt local backup,” type a password, then click Set Password.
To see the backups stored on your computer, choose Edit > Preferences, then click Devices. Encrypted backups have a lock icon in the list of backups.
Note: You can also connect iPhone to your computer wirelessly if you set up syncing over Wi-Fi.
This article explains and provides details about the different backup methods: Backup methods for iPhone, iPad, and iPod touch.
Afterwards, you can restore the backup by following the steps here: Restore your iPhone, iPad, or iPod touch from a backup.
Hope this helps. Take care.
