In BigQuery Web UI, there is a “Saved Queries” Section.
Is there way to access (read/write) those programmatically?
Any API?
Nope. It's a UI-only feature with a private interface.
Update 2021: We recommend using scripts to save and share queries.
Update 2025: The feature referenced in this question is "Classic saved queries", which are deprecated. See the following documentation for info about Dataform-based saved queries now available in BigQuery Studio.
https://cloud.google.com/bigquery/docs/saved-queries-introduction
You can also use routines (scripts and stored procedures) as a way to share SQL code assets between users.