Not necessarily. Most ordinary lenses transmit very little UV, but quartz is not the only option for UV work. Some enlarging lenses and even a few modern pancake lenses can transmit UV well enough to be useful.

If your goal is a true UV image, lens transmission matters a lot: a lens that blocks UV will limit what reaches the sensor no matter how the camera was converted. If your goal is a “full-spectrum” image in sunlight, UV usually contributes very little compared with visible and near-IR, so a lens blocking UV may make little practical difference to the final image.

To test a lens, measure spectral transmission: record the spectrum of a suitable light source with a spectroradiometer, then measure again with the lens placed between the source and the instrument. Dividing the two readings gives transmission versus wavelength. Make sure the light source has enough output in the UV range you care about.

So: no, quartz is not always required, but many standard glass lenses are poor for deep-UV work, and testing or published transmission data is the reliable way to know.