No, you can’t. Norton security software like Norton AntiVirus or Norton 360 is not supported on Amazon Fire OS based tablets (Kindle Fire or Fire).
The operating system of Kindle Fire tablets is Android based, but customized. The OS is called Fire OS. It is based on the Android Open Source Project, developed by Amazon for their devices.
Claims exist that many Android apps, including Norton apps, after some OS modifications can be installed onto Amazon tablet devices. Norton does not support this approach.