Taxi transfer

Google: Radio taxi Murcia. There aren't usually specific taxi companies but a company of taxis that you call, someone answers and send a taxi to your location. I am talking the word "taxi" as it's what you're asking for. A particular taxi driver can hand his business card to you if so you wish. 

That said, there should be taxis at airport exit happy to have your business. 

Radio taxi Murcia Tel. 968248800

You can call them to set up your pick up time to be on the safe side. 

Best wishes!