We observed the accretion-driven millisecond X-ray pulsars XTE J0929 - 314 and XTE J1751 - 305 in their quiescent states using Chandra. From XTE J0929 - 314 we detected 22 source photons ( in the energy range 0.3 - 8 keV) in similar to24.4 ks, resulting in a background- corrected time-averaged count rate of (9 x 2) x 10(-4) counts s(-1). The small number of photons detected did not allow for a detailed spectral analysis of the quiescent spectrum, but we can demonstrate that the spectrum is harder than simple thermal emission, which is what is usually presumed to arise from a cooling neutron star that has been heated during the outbursts. Assuming a power-law model for the time-averaged ( averaged over the whole observation) X-ray spectrum, we obtain a power-law index of 1,8(-0.5)(+0.6) and an unabsorbed X-ray flux of 6(-2)(+4) x 10(-15) ergs s(-1) cm(-2) ( for the energy range 0.5 - 10 keV), resulting in a 0.5 - 10 keV X-ray luminosity of 7(-2)(+5) x 10(31) (d/10 kpc)(2) ergs s(-1), with d the distance toward the source in kiloparsecs. No thermal component could be detected; such a component contributed at most 30% to the 0.5 - 10 keV flux. Variability in the count rate of XTE J0929 - 314 was observed at the 95% confidence level. We did not conclusively detect XTE J1751 - 305 in our similar to43 ks observation, with 0.5 - 10 keV flux upper limits between 0.2 and 2.7 x 10(-14) ergs s(-1) cm(-2) depending on assumed spectral shape, resulting in 0.5 - 10 keV luminosity upper limits of (0.2-2) x 10(32) (d/8 kpc)(2) ergs s(-1). We compare our results with those obtained for other neutron star X-ray transients in their quiescent state, and in particular with the quiescent properties of SAX J1808.4 - 3658. Using simple accretion disk physics in combination with our measured quiescent luminosity of XTE J0929 - 314 and the luminosity upper limits of XTE J1751 - 305 and the known spin frequency of the neutron stars, we can constrain the magnetic field of the neutron stars in XTE J0929 - 314 and XTE J1751 - 305 to be less than 3 x 10(9) (d/10 kpc) and (3-7) x 10(8) (d/8 kpc) G (depending on assumed spectral shape of the quiescent spectrum), respectively.