Why do Boeing aircraft have serrated engine nacelles, but Airbus does not?
It doesn't appear to be a patent thing. The main work on the chevron (shaped nozzles) was done by NASA. Every design decision in an aircraft is a compromise and this was no different. The addition of chevrons for noise suppression leads to a (small) increase in specific fuel consumption and Airbus considered it not worth it. Flightglobal quotes Dougie Hunter, A350 chief engineer as saying:


  We didn’t get a noise advantage from chevrons, and there is a specific fuel consumption penalty, so we’ve not adopted them.


Airbus seems to have gone in the insulation and high bypass route for reducing noise. From the same article:


  Goodrich is to build the engine nacelle for the A350’s GEnx engine, with Airbus supplying the composite inlet, which incorporates a “zero splice” acoustic inner barrel to provide a new level of fan-noise reduction.


As an example of the acoustic treatment of nacelle in A320 family, Airbus says:


  The thrust reverser structure was acoustically treated along approximately
  half of its internal surface. It was clear that increasing this acoustic area would increase the acoustic sound absorption and hence decrease the overall measurable exhaust noise. The inner fixed structure was redesigned with increased acoustic area. In addition, the support struts and several fairings were also modified to include acoustic treatment. The result was an increase in the acoustic area of the thrust reverser by approximately 2 square meters (21.5 sq ft).




Airbus does use chevrons in some of the aircraft, like A321 along with nacelle acoustic improvements; however, the chevrons are available only in the inner (core) nozzle. The package is available as standard for CFM powered A321 aircraft and options are available for A320. The image below shows a A321 with chevron nozzles.



Image from pprune.org

According to Airbus:


  The noise reduction package is currently offered as the basic standard on all CFM powered Airbus A321 aircraft. It is also optional on all the other A320 Family aircraft.