There are two ways an alien probe might indicate its presence: optically or via radio waves. Technically radio waves are incorporated into optics, but by optics I'm referring to the visible part of the electromagnetic spectrum (red to violet light: 380 to 750 nm wavelengths).
Detection via Visible Light Waves
Visible light astronomy, could detect a moving object within the solar system if telescopes where pointing in the right direction. The movement of the object would be noticed against the background of distant stars that from our perspective appear stationary.
By observing the moving object over a period of time a course can be determined. If the course followed by the object orbits the Sun it would be assumed the object could be an asteroid or a comet. If the object followed a "weird" path that did not orbit the Sun, but moved from planet to planet, or something else, it would be noted as something peculiar.
Consider the consternation caused by Oumuamua in 2017 when it came very close to Earth. It was the first object from outside the solar system to enter the system that was noticed. There was speculation by some people that instead of being just a rock from somewhere else it might have been an alien probe with the appearance of a flat rock.
Now if an object was detected optically, in addition to its path being weird if it flashed in an unnatural way, something akin to a type of alien Morse code, or a repetitive sequence of prime numbers then someone might postulate the object might be of extraterrestrial origin.
The technology and means for optical detection existed in the 1800s.
Detection via Radio Waves
Regarding radio waves the availability of technology and people needs to be considered. World War 2, which occurred during the first half of the 1940s affected both.
The first detection of radio waves from an astronomical object was in 1932, by Karl Jansky who was with Bell Telephone Laboratories when he was investigating static that interfered with transatlantic shortwave voice communications.
Inspired by Jansky's work Grote Reber built a 9 m parabolic radio telescope in his back yard, in Wheaton Illinois, in 1937. As others have stated in their answers, to be able to detect anything one has to be using the correct radio frequencies. Reber's first two attempts to detect anything from the cosmos failed because he used frequencies of 3300 MHz and 900 MHz. He only started getting results when he used 160 MHz. He ended up completing a radio frequency map of the skies in 1941, which was extended in 1943. Until the end of World War 2, Reber was the only radio astronomer in the world.
People started to enter the field of radio astronomy after World War 2. Confirming radio emissions from the Sun was some of the first work undertaken, usually using repurposed military radar equipment from World War 2. Ruby Payne-Scott was the first female radio astronomer from 1946 to 1951.
Many of the world's iconic radio telescopes where not constructed until the late 1950s, or even later:
- Dwingeloo Radio Observatory, Netherlands, completed 1956
- Jodrell Bank, United Kingdom, established 1945, near Manchester.
The Lovell Telescope completed in 1957
- Mullard Radio Astronomy Observatory, United Kingdom, established
1945, near Cambridge.
- Parkes Observatory, Parkes, Australia, built 1961, operational 1963
- Molonglo Observatory Synthesis Telescope, Australia, 1965
- Goldstone Observatory, California, USA, established 1958.
- The now decommissioned Arecibo Observatory, Puerto Rico,
completed November 1963.
- University of Illinois Radio Telescope, 1957
- Green Bank Telescope, West Virginia, USA, October 1961.
- Pushchino Radio Astronomy Observatory, Russia, established 1956. Four 22 m radio telescopes completed in 1959.
- Nançay, France, inaugurated 1965
- Effelsberg 100-m Radio Telescope, Germany, inaugurated in 1973
So if you want to set your story in the 1950s, then the signal could have been received in either: the Netherlands (1956), the United Kingdom (1957), the Soviet Union (1959), USA at Goldstone (1958) or possibly the University of Illinois (1957).
The other thing is, just because a radio telescope exists doesn't mean it can receive all signals. Size matters, the bigger the better, particularly for very weak signals, as NASA discovered during the Apollo 13 emergency when its 22 m diameter dish at Honeysuckle Creek, near Canberra, Australia could not clearly and reliably receive communications from the stricken Apollo 13 craft. The Parkes radio telescope had to be quickly converted from a dish that just received radio signals to one that could send and receive.
A serendipitous signal like the Wow! signal, received in 1977, could be a good analogy for what might have happened.