There are a lot of online resources that cover this, a quick Google will point you in the right direction.
I tend to use JSON Web Tokens (JWT) for this using tymon/jwt-auth for the backend auth side, you can use this as part of the built in auth mechanism. Then on the Vue side websanova/vue-auth
“API Authentication in Laravel-Vue SPA using Jwt-auth” by Olu Udeh https://link.medium.com/BLyvkvIIrS might get you started.
Neil