Not a definitive answer, but a close upper bound.
The same paper that had the first published computer-generated image of the Mandelbrot set also includes an image of the Julia set (no idea if it's the first one too).
The paper is from precisely the same year Julia died, 1978!
The conference was held on June 5-9, at the State University of New York, Stony Brook, and the proceedings were published on 1981.
- Robert Brooks and Peter Matelski, The dynamics of 2-generator subgroups of PSL(2,C), in Irwin Kra (1 May 1981). Irwin Kra, ed. Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference