It depends on the app that's generating the notifications. Some apps like Whatsapp and Plume to name a few will give you the option to choose what color your LED will blink when getting a notification, but most apps won't give you this option (I believe the default is Deep Blue).
However, there are apps like LightFlow that allow you to customize the LED to most apps' notifications in your phone. It even gives you more options like blinking speed and cout, etc. HOWEVER, I haven't tried LightFlow on the Note 3. It might not work and I do remember it had issues with the infamous 'Talk Back' feature/bug in Samsung phones and I'm not sure if this has been fixed on the Note 3 (most apps that used the "TalkBack" feature caused the device to, well, talk back all the time, even if you disabled the feature...only by disabling the app that was using it did the talking stop).