The L204 dark cloud complex is a nearby filamentary structure in Ophiuchus North that has no signs of active star formation. Past studies show that L204 is interacting with the nearby runaway O star, {zeta} Oph, and hosts a magnetic field that is coherent across parsec-length scales. Near-infrared H-band (1.6 {mu}m) linear polarization measurements were obtained for 3896 background stars across a 1{deg}x1.5{deg} region centered on the dense Cloud 3 in L204, using the Mimir near-infrared instrument on the 1.8 m Perkins Telescope. Analysis of these observations reveals both large-scale properties and small-scale changes in the magnetic field direction in Cloud 3. In the northern and western {zeta} Oph facing regions of the cloud, the magnetic field appears to be pushed up against the face of the cloud. This may indicate that the UV flux from {zeta} Oph has compressed the magnetic field on the western edge of L204. The plane-of-sky magnetic field strength is estimated to be ~11-26 {mu}G using the Chandrasekhar-Fermi method. The polarimetry data also reveal that the polarization efficiency (PE {equiv} P_H_/A_V_) steadily decreases with distance from {zeta} Oph (-0.09%+/-0.03%/mag/pc). Additionally, power-law fits of PE versus A_V_ for localized samples of probe stars show steeper negative indices with distance from {zeta} Oph. Both findings highlight the importance of external illumination, here from {zeta} Oph, in aligning dust grains to embedded magnetic fields.