How to implement this"Laravel+algolia+vuejs "

You can use Laravel Scout, it works with Algolia.

Currently, Scout ships with Algolia, Meilisearch, Typesense, and MySQL / PostgreSQL (database) drivers.

https://laravel.com/docs/11.x/scout#algolia

Algolia has some documentation on how to set this up with Vue https://www.algolia.com/doc/framework-integration/laravel/tutorials/getting-started-with-laravel-scout-vuejs/?client=php