Yes, never knew that, but the L4 and L5 positions for the Moon are not actually stable, due to perturbing influence of the sun. 1960BAICz..11..130S Page 130
So the Moon can't have permanent trojans. But given that it could have dust in the Kordylewski clouds and that we wouldn't expect to see an object there even up to a meter in diameter - it could have temporary trojans . The dust would be temporary trojans in a way, tiny trojans, so if it exists, then it is just a matter then of how large the trojans can be.
Our Earth does get temporary tiny moons such as 2006 RH120, 2-3 meters across which was a satellite of the Earth for nearly a year: September 2006 to June 2007. These temporary satellites spend most of their time in heliocentric orbit, but from time to time get captured by the Earth for a short time, months, or even could be for centuries, but eventually wander off into a separate heliocentric orbit again.
Based on that observation and models, someone worked, out, our Earth probably has one "moon" of about 1 meter across at any time, and it probably has about 7oo or so tiny moons at least 10 cm in diameter at any given time.
"They orbit at distances between five and 10 times as far from Earth as the moon. Most stay in orbit less than a year, although some stay much longer. One object in the team's simulations stayed in orbit for almost 900 years."