I also don't think there's a specification written for the jq query language, unlike https://jmespath.org/ , which as you mentioned also has more client libraries.
I too am probably going to embed jmespath in my app.I need it to allow users to fill CLI flags from config files, and it'll replace my crappy homegrown version ( https://github.com/bbkane/warg/blob/740663eeeb5e87c9225fb627... )