How do I permanently remove Messages app from MacBook Air?

You can't as Messages is part of the macOS suite of installed apps.

However, you can minimize it, by following these steps:

• First, sign out of iMessage:
  • Open Messages on your Mac.
  • Click Messages in the menu bar, then choose Settings (or Preferences).
  • Go to the iMessage tab.
  • Click Sign Out next to your Apple Account.
• Next, prevent Messages from Opening automatically:
  • Open System Settings > General > Login Items.
  • Look for Messages under “Open at Login” and remove it if listed.
• Next, block getting notifications from messages:
  • Go to System Settings > Notifications.
  • Select Messages and toggle Allow Notifications off.
• Finally, delete the Messages app from the Dock:
  • Right-click the Messages icon in the Dock and select Options > Remove from Dock.

You cannot remove the messages app

In fact, the Messages app has been installed on your Mac since you got it. It's been included with the macOS for over a decade.