Aiven PostgreSQL database's major version can be upgraded by forking the database service. Currently supported PostgreSQL major versions are 9.5, 9.6 and 10.
Navigate to the service's "Overview" page and you will see a "New database fork" button in the top-right corner. This will allow you to create a separate new database service that is cloned from the current one's backups.
When creating the fork, you can choose to upgrade to a more recent major version. The database automatically runs upgrade on itself at the end of the restoration providing you with a new PostgreSQL database service with the major version upgraded.
Stopping writes to the database, creating the fork and redirecting the traffic to the forked service will cause some downtime to your database service. We're working on improving this and have a roadmap item scheduled for Q2/2018 to be able to do in-place upgrades and thus drastically reducing the downtime required.