The answer is ...
YES, because I had the idea for this problem in the winter 1970 / 1971 when I was 17 years old.
NO, because nothing was published about this problem before 1975.
Probably, because it seems unlikely that no-one had had this idea in preceding decades or centuries.
Important note: Because I did not publish any result, I claim nothing and fully respect that Buckley is considered as inventor of the problem and that Scott Kim (and others) solved it.
It is sufficient for me that Martin Gardner published 4 of my tetrads (including the solution with order-11 polyominoes) in his book. 
My own Tetrad story
At the end of 1970, when I was 17 years old, I started to write recreational mathematical puzzles in a school notebook, and the corresponding solutions in another notebook.
I wrote down puzzles created by myself and others, (for example puzzles by Prof. Heinz Haber, that were shown on German TV in the 1960s.
The puzzle 22 is a dissection puzzle of the following type:
'Given is the shape of a piece of land. Dissect it into four congruent parts.'
Often another constraint was added. In this case each part has to have one tree.
Add the constraint that each of the parts has to have at least one edge in common with each of the other parts.
I searched for a suitable shape and found the following.
Today we would call the solution a 'tetrad'.
Of course I was not happy with the lake in the land of puzzle 34, and wanted to find such a dissection puzzle with no lake.
Solution of Puzzle 51
The left solution was my first 'tetrad' with no hole. Unfortunately I crossed out this artwork when I found the polyomino solution on the right.
With the polyomino solution I added the the constraint to puzzle 51 that among the four parts no mirror image is allowed.
This excluded the original solution and I marked it as wrong ('falsch').
Although the left tetrad looks rather complicated I like it, because it was my first solution. And this was important enough for me that I remember rather clearly how I found it. See question 6.
A few days or weeks later I found a 'tetrad' made of order-12 polyominoes.
In 1973 I started to study mathematics and physics at University of Erlangen. When I told professor about the 'tetrad' problem they had no time to search for their own solutions and wanted to see one solution immediately. But as soon as I drew the order-12 polyomino 'tetrad' on a blackboard they thought that the problem was too easy and of no importance. This negative experience caused me to abandon the tetrad problem for some years.
In 1978, the first edition of "Spektrum der Wissenschaft" a German translation of "Scientific American" appeared. I was interested in Gardner's articles, and searched for previous articles in a library. There I found the article with Scott Kim's tetrads. I was very pleased to see that the problem received attention. The polyamond and polyhex solutions were so nice that I wondered why I hadn't had the idea of using triangular or hexagonal grids.
From this time on I occasionally searched for a tetrad made of order-11 polyominoes, but always had in mind that such a tetrad most likely did not exist.
Surprisingly I was successful in 1978, or 1979, and sent this and some other tetrads to Martin Gardner. See question 9.
In the chapters (questions) 5 - 8 you can read how I found the tetrads. Website with tetrads
In March 1996 I presented a tetrad puzzle on my AOL homepage. This site was at least available until February 2001.
Contrary to the replies to my magic square puzzle I did not get any feedback on my tetrad puzzle.
Computerprograms for searching polyomino tetrads
In August 1998 I wrote a few 16-bit Basic programs. One program searches all polyominoes of a certain order n. Another program searches for polyominoes which can form a tetrad. And a third program draws the tetrads as BMP-pictures. This was done up to order n=13: here is a list of all 11300 suitable polyominoes for n=13 (meaning of the data: number of rows, number of columns, binary code for row 1, binary code for row 2, ..., 0 = end of polyomino). List of bitmaps for n=11. Main results: no holeless tetrads for n<11, only one holeless tetrad for n=11 and n=13. Nothing was published.
George Sicherman  also wrote computer programs for his research and Juris Čerņenoks could do this up to order n=17 and published it 2018.