Matter doesn't need internet access, it's an entirely local protocol even when you integrate to other vendor ecosystems.
Now, something like Google Home might decide to go online to talk to a Google Home Hub device, which is where Google wants to initiate all Matter communication from, but that's a Google thing, not a Matter thing.
Technically Matter itself doesn't require Internet access, but you'll come across many devices that won't operate (or even join) without a functioning border router. What's in the spec and the lived experience are sort of different here.
A border router does not provide internet connectivity.
A border router is used to bridge e.g. Thread networks onto a regular IP network so that your non-Thread smartphone can talk to it, and so Matter devices on different network types can link up directly if you want a Thread switch to control a WiFi light for example.
Matter devices use link-local IPv6 addresses and cannot route packets onto the internet even if they wanted to.
Now, something like Google Home might decide to go online to talk to a Google Home Hub device, which is where Google wants to initiate all Matter communication from, but that's a Google thing, not a Matter thing.