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We present seven epochs of spectroscopy on the quadruply imaged quasar SDSS J1004+4112, spanning observed-frame time delays from 1 to 322 days. The spectra reveal differences in the emission lines between the lensed images. Specifically, component A showed a strong enhancement in the blue wings of several high-ionization lines relative to component B, which lasted at least 28 days (observed frame) and then faded. Since the predicted time delay between A and B is ~30 days, our time coverage suggests that the event was not intrinsic to the quasar. We attribute these variations to microlensing of part of the broad emission line region of the quasar, apparently resolving structure in the source plane on a scale of [approx]10 super(16) cm at z = 1.734. In addition, we observed smaller differences in the emission-line profiles between components A and B that persisted throughout the time span, which may also be due to microlensing or millilensing. Further spectroscopic monitoring of this system holds considerable promise for resolving the structure of the broad emission line region in quasars.