We have carried out a high-resolution imaging search for the galaxy associated with the damped Lyman- \alpha ( DLA ) absorber at z = 1.859 toward the z _ { em } = 2.48 quasar QSO 1244+3443 , using the Hubble Space Telescope ( HST ) and the Near Infrared Camera and Multiobject Spectrometer ( NICMOS ) . Images were obtained in the broad filter F160W and the narrow filter F187N with camera 2 on NICMOS with the goal of detecting the rest-frame optical continuum and the H- \alpha line emission from the DLA . After PSF subtraction , two weak features are seen at projected separations of 0.16-0.24 ” from the quasar . Parts of these features may be associated with the DLA absorber , although we can not completely rule out that they could be artifacts of the point spread function ( PSF ) . If associated with the DLA , the objects would be \approx 1 - 2 h _ { 70 } ^ { -1 } kpc in size with integrated flux densities of 2.5 and 3.3 \mu Jy in the F160W filter , implying luminosities at \lambda _ { central } = 5600 Å in the DLA rest frame of 4.4 - 5.9 \times 10 ^ { 9 } h _ { 70 } ^ { -2 } L _ { \odot } at z = 1.86 , for q _ { 0 } = 0.5 . However , no significant H- \alpha line emission is seen from these objects , suggesting low star formation rates ( SFRs ) . Our 3 \sigma upper limit on the SFR in the DLA is 1.3 h _ { 70 } ^ { -2 } M _ { \odot } yr ^ { -1 } for q _ { 0 } = 0.5 ( 2.4 h _ { 70 } ^ { -2 } M _ { \odot } yr ^ { -1 } for q _ { 0 } = 0.1 ) . This together with our earlier result for LBQS1210+1731 mark a significant improvement over previous constraints on the star formation rates of DLAs . Dust within the DLA could extinguish H- \alpha emission , but this would require the dust content in the DLA to be much higher than that inferred from previous DLA observations . A combination of low star formation rate and some dust extinction is likely to be responsible for the lack of H- \alpha emission . Alternatively , the objects , if real , may be associated with the host galaxy of the quasar rather than with the DLA absorber . In any case , our observations suggest that the DLA is not a large bright proto-disk , but a compact object or a low-surface brightness galaxy . If the two features are PSF artifacts or associated with the quasar host , then the constraints on the size and luminosity of the DLA are even more severe .