Use SDIO pheripheral for quad SPI flash memory
No, you cannot.

While most SD cards do also support an SPI operating mode, the converse is not true; the SDIO peripheral on the STM32F103 only supports the SD protocol. It cannot communicate with SPI (or QSPI) devices.

The STM32F103 doesn't support QSPI. You'll need to communicate with your flash device using single-lane SPI, or use another device.