If I install syncthing on my Synology NAS which is connected to my home network and then install it on my laptop and sync the two within my home network, will my laptop sync to the NAS when I am outside my home network?
If not, are there any specific configurations to do with syncthing on the NAS or make any changes to my home router?
Yes. Use the default settings, and Syncthing should connect no matter how separated they might be. Syncthing prefers direct connection whoever that is possible and will try to get direct connection even though they are behind NAT. In case direct connection is not possible, it uses relays instead.
It might help to open some ports in your router to the outside. Either by enabling UPnP and having Syncthing use that to open it automatically, or by manually adding the respective entries in your router setup interface. I do the latter with my Synology NAS and it works quite well (with IPv6) although I am behind a DS-Lite tunnel from my service provider. When the laptop doesn’t have IPv6 connectivity in some remote network, it still works but uses relays.
Relays are shares resources. The connection speed when connected via relay is most often slower than if direct connection is possible. So: If speed is an issue, dig deeper. Otherwise you should be fine anyway.