Well, this question sounds a bit strange because these two are for different layers of your app. NodeJS is a backend - it can communicate directly with operating system, manage HTTP-requests, store data in database and so on. This can be useful for your task. Vue.js is a frontend framework - a library that makes it easier to create user interface for web apps. To use Vue.js you still need some sort of backend or webserver which will deliver Vue application to the browser.
So, I guess, you’ll need both of them 