As you said you can get it with props, which I prefer if I'm going to restrict that page to only visitors with tokens.
In Vue, if you don't have a liberary to get it, then yes you need to use window.location.search.
new URLSearchParams(window.location.search).get("page")
That has been mentioned by the author of Inertia: https://github.com/inertiajs/inertia-laravel/issues/174#issuecomment-716484179