Thank you for reaching out on Italiapass!
You can certainly take the train from Nice to Cinque Terre. Unfortunately italiarail.com8 is not offering this service at the moment. However, you can still book this trip on more generic european travel sites such as rome2rio.com8 or omio.com8.
If you have any questions during your booking process, feel free to contact us directly at 1 877 375 7245 or firstname.lastname@example.org.