Site search using Contentful with NextJS

HI there, I am new to Contenful, but just investigating a project for a client who has asked if they could use the Programmable Google Search on their site which is to be built using NextJS and Contentful.

I have seen it is possible to use Contentful’s inbuilt search, or a third party like Algolia. I am wondering if anyone has had any experience with Google Site Search and Contentful?