Using Google Ngrams to search for the phrase "Publish or Perish", we see that the phrase seems to have entered reasonably widespread use in the late 1940s and peaked in the mid 1960s:
Searching Google Books for the phrase turns up a number of earlier uses, but many seem to be mis-identified by Google (e.g. they suggest that Nature vol 362 is from 1869, whereas it was actually published in 1993).
However, Google Books did point me to this, published in 1935 in the Transactions and Proceedings of the Royal Society of New Zealand:
One of the aphorisms of Professor W. M. Davis, whose death was referred to at an earlier meeting to-day, was “Publish or perish.” Davis applied the remark to individuals, for he went on to say, “If it is worth doing, it is worth printing!” If no opportunity to print is afforded, then the well of inspiration dries up. It may also be applied to societies, for this generally disconsidered function of such a society as this is really the main stimulus to a vigorous existence.
So even if the phrase wasn't invented by Davis, it was in use, without needing detailed explanation, in the 1930s; but was still considered a relatively new concept at the time, since it could reasonably be ascribed as a quote from a single individual.
(Also, though it's hard to tell, this early use seems to be more positive than we're used to seeing.)
Wikipedia claims that it first occurred "in an academic context" in 1928, which fits with this timeline.