Yes. EPCQ-A devices are supported in the Intel® Quartus® Prime software version 17.1 and later, but Cyclone® III and Stratix® III are not supported in those versions. When you need to modify your design for Cyclone III or Stratix III device to migrate the configuration device to a Quad-Serial Configuration (EPCQ-A) devices, you can use the Quartus II version 13.1.4 and a patch that enables the IPs to support EPCQ-A devices. When the version you use is earlier than 13.1.4, upgrade to 13.1.4 and apply the patch.
You can download the patch from FPGA Configuration Devices Support > Software Download.
To check if your design is compatible with EPCQ-A devices, see AN822: Intel FPGA Configuration Device Migration Guideline.