Probably? I just have it setup to always be running in the background on my devices. So if it detects a file change to my sync folder that gets sent to all other devices currently connected. I use global sync so as long as the device has an internet connection or is on the same local network it should be able to sync.

There is an API to interface with syncthing daemon running on your computer https://docs.syncthing.net/dev/rest.html so if you wrote a program to track the nfc action and interface with that API I think you probably could.