Hi Jason,
Please carefully follow my instructions below to create a new Service and subscription limits (quotas) request to increase your quota to deploy Azure Database for PostgreSQL Flexible Server to higher tier.
1. Navigate to this link to start the process:
https://portal.azure.com/#create/Microsoft.Support
2. Enter quota in the box and click Go
3. Select Others / Service and subscription limits (quotas) and click Next
4. Click Create a support request button, as shown
5. Select Issue type: Service and subscription limits (quotas), Subscription, Quota type: Azure Database for PostgreSQL Flexible Server, click Next
6. Now you should be on Additional details tab. Click Enter details
7. Select Quota increase, select Location you need, select Series you need, enter number of cores needed, click Save and continue
8. Finish filling out Additional details tab, click Next, review and click Create. You should receive an automated email within a few minutes.
NOTE: It is possible that they will deny your request due to capacity limit or other reason, but at least you tried.
Please click Accept Answer and upvote if the above was helpful.
Thanks.
-TP