To begin, Type-C connection standard explicitly prohibits the use of proprietary power delivery protocols such as QC. Since USB Type-C spec release 1.3:
184.108.40.206 Proprietary Power Source
This section has been deprecated. Devices with USB Type-C connectors shall only employ signaling methods defined in USB specifications
So the formal answer to first part of your question, no, "no USB Type-C cables support QC method". However, since QC uses USB2 wires, and every Type-C cable does have USB2 wires, every USB C-C cable can support QC-whatever, if someone is willing to violate USB Type-C specifications in their product.
Regarding the second part of this question, the cable communication channel is just a wire inside the cable, between two ends. So whatever PD mode is implemented in host and agreed by the device (link partner), it will be supported/negotiated. Again, SOP processors that are (must be) embedded into every C-C cable, will inform the host about cable capabilities, and PD firmware will restrict the power delivery accordingly. So technically you personally don't need to check anything, everything will be checked out by PD firmware in the USB host port.
If you mean by "checking" as a person developing the PD firmware for a USB host, you should know better.