We present spectroscopic measurements of the [O III] emission line from two subregions of strong Ly alpha emission in a radio-quiet Lyman-alpha blob (LAB). The blob under study is LAB1 at z similar to 3.1, and the [OIII] detections are from the two Lyman break galaxies (LBGs) embedded in the blob halo. The [OIII] measurements were made with LUCIFER on the 8.4 m Large Binocular Telescope and NIRSPEC on 10 m Keck Telescope. Comparing the redshift of the [OIII] measurements to Ly alpha redshifts from SAURON allows us to take a step toward understanding the kinematics of the gas in the blob. Using both LUCIFER and NIRSPEC we find velocity offsets between the [OIII] and Lya redshifts that are modestly negative or consistent with 0 km s(-1) in both subregions studied (ranging from -72+/-42-+6+/-33 km s(-1)). A negative offset means Ly alpha is blueshifted with respect to [O III] a positive offset then implies Lya is redshifted with respect to [O III]. These results may imply that outflows are not primarily responsible for Lyman alpha escape in this LAB, since outflows are generally expected to produce a positive velocity offset. In addition, we present an [O III] line flux upper limit on a third region of LAB1, a region that is unassociated with any underlying galaxy. We find that the [OIII] upper limit from the galaxy-unassociated region of the blob is at least 1.4-2.5 times fainter than the [OIII] flux from one of the LBG-associated regions and has an [O III] to Ly alpha ratio measured at least 1.9-3.4 times smaller than the same ratio measured from one of the LBGs.