All of the installation hardware to which that internal Zip drive is attached can't be used in any Apple computer that had an optional Zip drive. This includes the beige G3s, B&W G3s, and G4s. Apple always designed its own mounting sleds/carriers, that were tailored to their specific chassis builds. PCs have used a standardized form factor for some time, which eliminates the headaches often encountered when working with older Apple computers. In the photos, the bottom piece of steel is a mounting sled taken from a PC chassis, and these did vary from case to case. Attached to that sled is the Iomega 5.25" carrier with bezel, for installing a Zip drive in a PC's optical drive bay. If you were installing the Zip drive in a 3.5" bay, you'd remove the drive from the carrier and install it directly. This is what you'll be doing in your G4, because your Quicksilver has a drive carrier in which the optical drive and (optional) Zip drive are installed. This carrier must be removed from the computer chassis, in order to remove the Zip drive carrier from the underside of the optical drive carrier. The (4) mounting screws that attach the carrier to the chassis must be removed, in order to slide the carrier back and out, so you can disconnect the ribbon cable and power cable connections. The optical drive's ribbon cable has a piece of silver tape, that is stuck to the top of the drive carrier, that you must carefully remove for reuse. With the drive carrier out of the computer, the Zip carrier slides back a little to disengage it from the underside of the optical drive carrier. Make sure that the configuration jumper on the back of the Zip drive is set for "slave," which is Iomega's default setting. The Zip drive itself is installed in/removed from the carrier from the front and is secured with (4) machine screws. Place the Zip carrier on the underside of the optical drive carrier and slide it forward to lock it in place. Installation of the drive carrier is the reverse of the removal procedure. If your Quicksilver originally had a Zip drive installed, there would be a grey Zip bezel (with a slot cut out for disk insertion) installed on the face of your computer, below the optical drive door. If your Quicksilver's face has a blank cover installed there, it's unlikely that a Zip drive was ever installed. Either type of bezel is removed from the face of the computer by pressing in on the bezel's release tab (right side of bezel, located just inside the computer chassis, near the Zip carrier) and swinging it out, until the left side of the bezel unhooks. You can use the Zip drive without the Quicksilver Zip bezel - it just gives it a finished appearance. Used bezels can still be found at eBay, but whatever you do, don't buy a new one from "We Love Macs Money." They seem to think that scarcity is justification to charge prices that always exceed the market value of the target computer, into which the small part is to be installed.
Keep in mind that Zip drives had a design flaw that caused what has been called "click death," and this defect ruined disks and made the drive unusable. From what I read, disks that were damaged in a problematic drive could damage the read/write heads, when inserted into a functioning drive. This problem seemed to be more common with the Zip 100 drives than the Zip 250 models, but don't count on that. If I were you, I'd prefer to purchase a new (old stock) drive, because you never know how much service life is left in a used drive. And given the effort required for installation of an internal drive, I'd again recommend purchasing an external USB Zip 250 drive.