I believe PPC Restitution in Spokane sends out civil demand letters, so what you paid was a civil fine, not a criminal one. So long as you were not arrested, charged with or convicted of an actual crime, this incident should not appear on a background check.
The information provided in this answer does not create an attorney-client relationship and is not considered to be legal advice.
I agree with the above answer. Also if you have internet access and $10 your can run your own report on WSP WATCH located here. https://fortress.wa.gov/wsp/watch/
Unless you were charged with a crime, and it sounds like you were civilly assessed restitution, this offense will no appear on a criminal background check.