Legal Question in Civil Litigation in California

Motions to quash are used to either challenge defects in service or a state court's lack of jurisdiction. There is also a special motion to quash in family law proceedings which does not apply to your case.

The big concern that I immediately see is that it appears from your post is that it appears that the procedure for fee arbitration has not been properly followed by the attorney. An arbitrator's award in fee arbitration (MFAA) is not binding, unless the parties agree otherwise in writing after the dispute has arisen. If the attorney did not agree with the award, he must petition the court for a trial de novo within 30 days after notice of the arbitrator's award. (Bus. & Prof. Code, sect. 6204 subd. (a).)

I don't know where you are in this time frame.

Motions are rarely used in small claims, and one of the questions an attorney is going to want to know in detail is how you were aware of the small claims demand and what manner of service the proof of service. In the end, a motion to quash may just be a waste of your time, which would be better spent preparing a defense on the merits. It appears to me that you had a fee arbitration award in your favor, which is a good sign.