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On Wikipedia it is written that
In the 1950s, ROC curves were employed in psychophysics to assess human (and occasionally non-human animal) detection of weak signals.
Green and Swets (1966) is cited as attesting to that fact. I don't have any way of accessing that book, and nor would it necessarily constitute the first proposal of ROC curves for usage in psychology.
Green, D. M., & Swets, J. A. (1966). Signal detection theory and psychophysics. Society, 1, 521.
I worked out the answer to the question just as I was about to post it.
Swets (1973) writes of a 1954 conference that
Peterson and Birdsall had presented it a year earlier in a technical report. So it is fair to say, from the vantage point of psychology, that Peterson and Birdsall showed us how to plot the data.
Unless Swets is mistaken, the answer is Peterson and Birdsall (1953).
Peterson, W. W., & Birdsall, T. G. The Theory of Signal Detectability (1953). Technical report No. 13, Electronic Defense Group, University of Michigan, Ann Arbor.
Swets, J. A. (1973). The relative operating characteristic in psychology. Science, 182(4116), 990-1000.