Yes, they investigated UFOs during the Cold War because it was the COLD WAR and they wanted to see if Russians were sending spyplane.

Project Blue Book was Cold War, but military interest in unidentified flying objects and thus public interest was earlier, back to World War II.

https://en.wikipedia.org/wiki/Foo_fighter

Royal Air Force personnel had reported seeing lights following their aircraft from as early as March 1942,[11][12] with similar sightings involving RAF bomber crews over the Balkans starting in April 1944.[13] American sightings were first recorded by crews from the 422nd Night-Fighter Squadron stationed in Occupied Belgium during the first week of October 1944. At the time, these were erroneously believed to be Messerschmitt Me 163 rocket-powered interceptors, which did not operate at night.[14] However, the bulk of the sightings started occurring in the last week of November 1944, when pilots flying over Western Europe by night reported seeing fast-moving round glowing objects following their aircraft. The objects were variously described as fiery, and glowing red, white, or orange. Some pilots described them as resembling Christmas-tree lights and reported that they seemed to toy with the aircraft, making wild turns before simply vanishing. Pilots and aircrew reported that the objects flew together in formation with their aircraft and behaved as if they were under intelligent control, but never displayed hostile behavior. However, they could not be outmaneuvered or shot down. The phenomenon was so widespread that the lights earned a name – in the European Theater of Operations they were often called “Kraut fireballs”, but for the most part called “foo fighters”. The military took the sightings seriously, suspecting that the mysterious sightings might be secret German weapons, but further investigation revealed that German and Japanese pilots had reported similar sightings.[15]

On 13 December 1944, the Supreme Headquarters Allied Expeditionary Force in Paris issued a press release, which was printed in The New York Times the next day, officially describing the phenomenon as a “new German weapon”.[16] Follow-up stories, using the term “Foo Fighters”, appeared in the New York Herald Tribune and the British Daily Telegraph.[17]

In its 15 January 1945 edition, Time magazine carried a story titled “Foo-Fighter”, in which it reported that the “balls of fire” had been following USAAF night fighters for over a month, and that the pilots had named it the “foo-fighter”. According to Time, descriptions of the phenomena varied, but the pilots agreed that the mysterious lights followed their aircraft closely at high speed.[18]

The “balls of fire” phenomenon reported from the Pacific Theater of Operations differed somewhat from the foo fighters reported from Europe; the “ball of fire” resembled a large burning sphere that “just hung in the sky”, though it was reported to sometimes follow aircraft. There was speculation that the phenomena could be related to the Japanese fire balloon campaign. As in Europe, no aircraft were reported as having been attacked by a “ball of fire”.[19]

The postwar Robertson Panel cited foo fighter reports, noting that their behavior did not appear to be threatening, and mentioned possible explanations, for instance that they were electrostatic phenomena similar to St. Elmo’s fire, electromagnetic phenomena, or simply reflections of light from ice crystals. The Panel’s report suggested that “If the term ‘flying saucers’ had been popular in 1943–1945, these objects would have been so labeled.”[2]

You then had the concern over the Japanese fire balloons:

https://en.wikipedia.org/wiki/Fu-Go_balloon_bomb

And then, the year after the war ended, the flying saucer craze:

https://en.wikipedia.org/wiki/1947_flying_disc_craze