Yes! But you need a paying account.
If you want to use a Postgres server run by us:
- If you have a free account, go to the Account page, and click on one of the buttons for upgrading to a paid account. In the dialog that appears, switch the Postgres option on, choose how much disk space you want for your Postgres databases, choose any other paid features you want, and then click "Upgrade to this custom plan". This will take you through a payment confirmation process.
- If you already have paying account, go to the Account page, and click on the "Customize your plan" button. The popup window will be pre-configured with all of your current custom plan settings, so you just need to switch the Postgres option on, choose how much disk space you want for your PG databases, and then click "Switch to this custom plan". This will take you through the normal payment confirmation process. (You won't be charged the new price until your next billing date.)
Once that's done, go to the "Databases" tab, then click the "Postgres" button. You'll have a new option to create a Postgres server and specify the adminstrator password.
If you want to use a Postgres server run by some other service
Just upgrade to any paid PythonAnywhere plan -- the cheapest one will work just fine. You don't need to check the "Postgres" option -- that is for creating a PythonAnywhere-hosted Postgres server, which of course you don't need.
Paid PythonAnywhere accounts have unrestricted Internet access so once the upgrade has gone through, you will be able to connect to your remote Postgres instance.