The only big rust component was introduced a couple of releases ago: Stylo.
Once Webrender is in Firefox, a serious chunk of Firefox will be written in Rust.
So yes, Quantum is faster as a direct result both of Rust code, and of Rust's memory-safety-makes-parallelism-practical features. That is not the only source of performance improvement in Quantum though.
Also, Quantum isn't yet getting the full benefits possible from this code for a few reasons. Firefox 57 uses Stylo for content, but not yet for chrome, which will be coming in a later release. In Servo, CSS is parsed off the main thread, but in Quantum it is not yet (will be done in a future release). Servo pipelines style resolution and frame construction (basically after the top down pass to deal with the style cascade, we go back up the tree bottom up constructing the layout data structures), and Quantum does not yet do this. Lastly, cross-language inlining is missing which would allow inlining FFI calls. Servo doesn't have this issue since all the driver and layout code is also in Rust.
"Quantum" as a term covers a large number of areas, this blog post covers it well:
https://hacks.mozilla.org/2017/11/entering-the-quantum-era-h...