The common CH341A programmer boards cant use quad SPI mode for faster programming as the extra data I/O lines from the CH341A chip aren't connected to anything. It would be theoretically possible, *if* the right connections were present, but would need software support and would probably be incompatible with the use of the usual software and ordinary non-quad SPI FLASH chips in the same socket.
There is however a limitation of how the ZIF socket is wired, that's incompatible with certain SPI FLASH chips *IF* they have been configured for quad SPI mode, which uses those pins as I/O so naturally cant tolerate them being shorted to rails. Unfortunately the bit in question in the FLASH chip's status register is non-volatile so quad SPI mode is 'sticky', as opposed to the sane alternative of making that bit volatile and having the host processor configure the chip appropriately for the application circuit on startup. See
Winbond's W25Q64FV datasheet for details.
The proposed mod is simply to make three track cuts disconnecting those pins from the Vcc trace on the bottom side running under the ZIF socket, to add a wire link to patch the interrupted Vcc track back together, leaving out the /HOLD pin, then to fit pullup resistors from the Vcc track to the /HOLD and /WP pins. *IF* you need the capability to program DIP-8 package SPI FLASH chips with non-volatile quad SPI mode, and to be able to enable that mode with the chip in the ZIF socket you need to do the mod. Some magnet wire and a couple of 0402 10K resistors should let you do it very neatly. Any resistors bigger than 0603 will be problematic due to the limited space between the Vcc track and the /Wp pin. Otherwise, if you are programming in circuit, simply make sure your test clip lead includes 10K resistors in series with the /HOLD and /WP pins.