Cloudflare Workers doesn't offer a Postgres option. At most, there's Hyperdrive to connect to Postgres hosted elsewhere, and even then Hyperdrive is essentially a connection pooler, with all the additional restrictions that places on you.

It's also unlikely that Cloudflare will offer a true Postgres option; their whole ethos is that anything you deploy is deployed globally, whereas relational databases enforce having a single server as a definition of truth. But I'd love to be proven wrong here.