Prab_, Final Cut Pro X 10.2 as well as Motion 5.2 are super picky when it comes to installed fonts.
Most likely you have a font conflict.
There are several things you can do. I would do, in the following order:
1) Validate fonts. Go to the following support document, scroll down to Validate and check your fonts:
https://support.apple.com/en-us/HT201749
If problems are found, delete the affected fonts
If validating doesn't help there can still be font issues.
2) Check if the problem is bound to the current user.
Create a new user in System Preferences, log in as the new user and start FCP. If it works, the problem is with fonts in your usual user's Library. If not, it's with system fonts.
If FCP starts for the new user, log in as your usual user, go to your user's Library (in Finder type: Command-Shift-G, then: ~/Library) and move the folder called Fonts to the Desktop. Log out and back in and see if you can start FCP.
3) If that doesn't help or if the problem was System related, open Font Book, go to File and "Restore Standard Fonts". Log out and back in and test FCP.
Alternatively to restoring the standard fonts, you can clean and rebuild the font cache (that's what helped me).
To do so, open Terminal and use the following commands:
atsutil databases -removeUser
This will remove the current user's font cache.
If you want to delete the font cache for all users, use this command followed by giving your admin password:
sudo atsutil databases -remove
The password won't show up but will be accepted.
Now stop the ATS server:
atsutil server -shutdown
Then restart it:
atsutil server -ping
Hope something will help.