Sorry to have to do this, but the Colibri, a great poster, gets Google ngrams wrong.
Underneath the chart you should see “Search in Google Books” and a button for 1800-1982.
If you click on that button, a miscellany of supposed hits appear. I say supposed, because their search engine will bring up books that do not contain the desired phrase. In fact, the 1976 hit does not contain “nuke the whales.” The 1977 hit does, but it’s from a 1979 magazine. Google Books dates bound volumes of magazines, such as usually found in the university libraries they copied from, by the date of the first issue in the volume. Bound volumes can contain years worth of quarterly magazines. The same fate might have befallen one of the two 1978 hits, hidden behind snippet view; the other is missing the phrase.
The first one I confirmed is the August 29, 1979 New York magazine, in a competition for provocative bumper stickers. Others from that year are snippet hits from magazines so they may be from earlier in that year.
Google Books and Google Ngrams are a language you have to learn to translate into usable form. They’re both indispensable and maddening.
I had (well I still have but it doesn’t fit any more) a T-shirt that read “Nuke a godless communist gay baby seal for Christ” I did have to be careful where I wore it. I usually wore it in the winter under a hoodie or flannel shirt.
The earliest I remember seeing the phrase was in the ads in the back of Rolling Stone magazine. The ad was for a T-shirt emblazoned with the phrase. The ad also read “Keep 'em Guessing!” The only problem is I don’t remember if it was in my junior year study hall ('76-'77) or freshman year at college in the fraternity house ('78-'79).
Sorta related: There’s “Pave the Bay” which was probably coined by those who don’t care for environmental restrictions in response to the Chesapeake Bay Foundation’s motto “Save the Bay”.
The Fleshapoids record by that name (their only release per Discogs) is pretty lousy. I have a copy, and assure you the phrase already was current by then. It was a 45, but that doesn’t make it punk - anyone who bought it would have to agree; if I were to guess no such band ever played a gig in SF clubs, just a pickup group with some guy repeating “nuke the whales, drop a big one on their tails”, and rhymes of similar caliber. Good enough for Dr. Demento though - he must have been desperate.