Ran into a similar problem, AWS Team told me that if your database is sitting in a non-default VPC and is publically accessible, then you have to whitelist IP address in your security group based on your region of managed grafana.
Here is the list of ip addresses based on the region.
• us-east-1: 35.170.12.166 54.88.16.229 3.234.162.252 54.160.119.132
54.196.72.13 3.213.190.135 54.83.225.191 3.234.173.51 107.22.41.194
• eu-central-1: 18.185.12.232, 3.69.106.181, 52.29.127.210
• us-west-2: 44.230.70.68, 34.208.176.166, 35.82.14.62
• us-east-2: 18.116.131.87, 18.117.203.54
• eu-west-1: 52.30.158.152, 54.247.159.227, 54.170.69.237, 52.210.87.10,
54.73.6.128, b54.78.34.200, 54.216.218.40, 176.34.91.249, 34.246.52.247
• us-east-2: 35.170.12.166, 54.88.16.229, 3.234.162.252, 54.160.119.132,
54.196.72.13, 3.213.190.135, 54.83.225.191, 3.234.173.51, 107.22.41.194
You can refer the documentation provided by aws on how to connect to the database at:
AMG Postgresql Connection