I just updated my os to Mac OS Monterey , now I found one of my application which is Final Cut Pro x 10.4.6 does not work and needs to be updated so I try to update Final cut to 10.7.1 from App Store but I could not see the update icon for it

This has been an App Store issue with the Pro Apps for ever.

In the App Store, go to the FCPX product page, it'll let you update there, even though the update does not appear on the update page.

What OS is this? Move your current version of FCP to the trash. Do not empty it. Go into the App Store to your Purchased page. Find the application there, not in Update or the FCP page. You can download from your Purchased page the latest version of FCP available to your OS.