The surface resistance of copper samples with an amorphous carbon (a-C) coating or with laser surface structuring, the surface treatments of choice for electron cloud suppression in critical cryogenic sectors of the high-luminosity upgrade of the Large Hadron Collider (HL-LHC), has been measured for the first time at a cryogenic temperature using the quadrupole resonator at CERN. Three different frequencies of relevance for evaluating beam impedance effects, namely, 400, 800, and 1200 MHz, have been investigated. No significant increase in surface resistance is observed for the a-C coating, compared to plain copper. In the case of laser structuring, the surface resistance depends on the direction of the surface currents relative to the laser-engraved groove pattern. The increase is minimal for parallel patterns, but in the perpendicular case the surface resistance increases considerably. Radio frequency (rf) heating from wake losses would then also increase in the HL-LHC case; however, the reduction in the power deposited onto the cold surfaces thanks to electron cloud suppression would still outweigh this effect.