Hi Bea
Did a little research because the topic is of interest to me as well.
Apparently, there is a bus service from Cartagena picking up at all the towns along the N322 - this goes into Alicante and then you pick up the C6 at the bus station to take you out to the airport. It would appear there is no direct bus to Alicante Airport to Los Alcazares.
Here is a link to some information and timings etc. and hope this helps (and I hope it is accurate and if it is not then I am sure someone else will correct this).
http://www.torreviejainformation.com/bus-details/
Hope this helps
Sandra