Measuring the dispersion of photonic devices with small dispersion-length products is challenging due to the phase-sensitive and alignment-intensive nature of conventional methods. In this Letter, we demonstrate a quantum technique to extract the second- and third-order chromatic dispersion of a short single-mode fiber using a fiber-based quantum nonlinear interferometer. The interferometer consists of two cascaded fiber-based biphoton sources, with each source acting as a nonlinear beam splitter. A fiber under test is placed between these two sources and introduces a frequency-dependent phase that is imprinted on the biphoton spectrum (interferogram) at the output of the interferometer. This interferogram contains the dispersion properties of the test fiber. Our technique has three novel features: (1) the broadband nature of the biphoton sources used in our setup allows accurate dispersion measurements on test devices with small dispersion-length products; (2) our all-fiber common-path interferometer requires no beam alignment or phase stabilization; and (3) multiple phase-matching processes supported in our biphoton sources enable dispersion measurements at different wavelengths, which yields the third-order dispersion achieved for the first time, to the best of our knowledge, using a quantum optical technique.