You should better explain what you mean by "fully automatic". Usually a weapon is called "fully automatic" if it loads and shoots with bursts (a machine gun, for example). Unlike semi-automatic which only loads
automatically, like an automatic pistol.
Such fully automatic antiaircraft guns of course existed in WW II, and even in WW I. (First specialized antiaircraft weapons were machine guns).
But apparently you mean automatic aiming rather than shooting. This was performed by a special device which is called antiaircraft director which slowly evolved, beginnng from WW I, with most
important development happening exactly during WW II. It is necessary because in AA artillery there is no enough time to compute the gun direction "by hand".
At the time of WW II this was an analog computer
(rather than digital). The complexity and sophistication varied. But it performed all necessary functions, except detection of the targets.
All was connected in one system: a distance and direction measuring device, computer and the gun which was aimed by an electric motor connected with this computer. The new ingredient, radar (for detection and aiming, usually separate radars) were added in the
end of WW II, and became widespread immediately after.
With the advent of digital computer, the system very much improved, but the principle remains the same: a radar detects an airplane, then coordinates are measured
by another radar, or by an optical device, then the data is processed by a computer, which computes the
aiming data and sends them to the gun by wire. A system of detection whether the aircraft is an enemy aircraft or a friendly one is also included. And much more. (Computer takes into account the wind, and how it varies with altitude, air temperature and pressure, propelling charge temperature, wear of the barrel, etc. All this is necessary to have a good chance to hit a fast moving airplane.) All this existed at the time of WW II, or was developed during that war. A typical antiaircraft projectile had initial velocity less than 1000 m/sec. If an airplane flies 150 m/sec at a distance of few kilometers,
you imagine how difficult it is to compute the aiming of the gun. If it flies closer to you, it is even more difficult.
In night time the role of direction finding radar was played by a search light which was coupled to the computer. The operators were expected to catch the airplane with the light beam, and follow it. The data were processed with computer and the result sent to the gun directing motors by wire. Two coupled search lights gave you a good range finding, by solving a triangle on the computer.
But it is still not "fully automatic" in the sense that
you describe: the DECISION to shoot is usually made by a human:-) And in most cases the target selection, if there are several targets around.
Figure 7 in this page shows an analog antiaircraft director
of the type used in WW II next to the gun that it controls. By the way the gun shown in the picture is not automatic: it was loaded manually but directed automatically.
The big horizontal tube on top is the optical range finder. The big box is occupied by the analog computer. The device is operated by 3 people:
one continuously measures the range, two others altitude and azimuth.
All these functions were automatized with the advent of aiming radar which can do this automatically. This page (scroll down to second photo) shows a rare photo of Soviet-made director ПУАЗО-3 of 1939, it has no range finder. Here is a German one.