To qualitatively measure the band-gap, you may use a UV-Vis spectrophotometer. Here is an excellent resource that walks through the science step-by-step: https://archive.cnx.org/contents/8e8a1e1b-0d9f-453b-95e1-bc95782973da@2/band-gap-measurements-of-quantum-dots
Additionally, they provide a link to a publication with more resources:
UV-Vis spectroscopy is excellent technique, to measure the band gap. We use Beer–Lambert law to calculate the absorption coefficient. Then according to Tauc relationship, the band gap is related to the absorption coefficient (α).
To be able to determine the band gap of quantum dots materials, you also need to measure photoluminescence spectroscopy in addition to absorption spectroscopy as the above two researchers' mentioned. The details of theory you could read in the following link.