[en] The Galactic magnetar SGR 1935+2154 has been reported to produce the first example of a bright millisecond-duration radio burst (FRB 200428) similar to the cosmological population of fast radio bursts (FRBs). The detection of a coincident bright X-ray burst represents the first observed multiwavelength counterpart of an FRB. However, the search for similar emission at optical wavelengths has been hampered by the high inferred extinction on the line of sight. Here, we present results from the first search for second-timescale emission from the source at near-infrared (NIR) wavelengths using the Palomar Gattini-IR observing system in the J band, enabled by a novel detector readout mode that allows short exposure times of ≈0.84 s with 99.9% observing efficiency. With a total observing time of ≈12 hr (≈47,728 images) during its 2020 outburst, we place median 3σ limits on the second-timescale NIR fluence of ≲18 Jy ms (13.1 AB mag). The corresponding extinction-corrected limit is ≲125 Jy ms for an estimated extinction of A _J = 2.0 mag. Our observations were sensitive enough to easily detect an NIR counterpart of FRB 200428 if the NIR emission falls on the same power law as observed across its radio to X-ray spectrum. We report nondetection limits from epochs of four simultaneous X-ray bursts detected by the Insight-HXMT and NuSTAR telescopes during our observations. These limits provide the most stringent constraints to date on fluence of flares at ∼10¹⁴ Hz, and constrain the fluence ratio of the NIR emission to coincident X-ray bursts to R _NIR ≲ 0.025 (fluence index ≳0.35).