Can astronauts on the moon observe a meteor shower?

Basically, no. Their spacesuit helmets were darkened, as lunar sunlight is much brighter than even desert sunlight.

I believe some astronauts may have reported seeing stars in the top port of the LM. This would have been when they were inside, with the shades drawn to block out light from the surface, when they were trying to sleep.

The brightness of the surface meant that cameras had to be adjusted to let in very little light, so stars did not show in the shots. Ask any photographer how this works. Tell them the Apollo film was ASA 100, at 1/250 of a second, or thereabouts. But Apollo 16 took a special camera, shielded from bright light. It took almost 200 pictures of stars! So much for the people who believe Apollo was a hoax because stars weren't seen in pictures.

I think shows like Star Trek have made people think that once you are in space, stars are bright enough to be seen from inside a well-lit room. Sorry, no.