Can a single PostgreSQL query use multiple cores?

No, for versions of PostgreSQL prior to v9.6. Please see the PostgreSQL FAQ: How does PostgreSQL use CPU resources?

The PostgreSQL server is process-based (not threaded). Each database session connects to a single PostgreSQL operating system (OS) process. Multiple sessions are automatically spread across all available CPUs by the OS. The OS also uses CPUs to handle disk I/O and run other non-database tasks. Client applications can use threads, each of which connects to a separate database process.

Since version 9.6, portions of some queries can be run in parallel, in separate OS processes, allowing use of multiple CPU cores. Parallel queries are enabled by default in version 10 (max_parallel_workers_per_gather), with additional parallelism expected in future releases.

PostgreSQL 9.6+ onwards, would start to see Parallel-Query finally coming to PostgreSQL.

For e.g. Concepts like Parallel Scan / Parallel Join / Parallel Aggregates are now already baked in, with more to come soon.

What's really exciting is that there are reports confirming near-linear speed-up in some cases, which is pretty impressive!

No, but there is a workaround. :)

I found parsel (parallel select) plpgsql function, which splits your query based on primary key, then connects to the database via dblink extension and waits for all subqueries.

https://gist.github.com/mjgleaso/8031067

Author also wrote article about this function: http://geeohspatial.blogspot.com/2013/12/a-simple-function-for-parallel-queries_18.html