Is there any cooperation between South Korea and Japan in the area of defense production given they have several common enemies in the region?
Short answer: Yes, South Korea, Japan, and the United States participate in joint military exercises together
Long answer: South Korea certainly has an enemy in North Korea, and Japan has significant interest in preventing North Korea from engaging in hostilities because they could easily get caught in the line of fire, which is likely the cause of the current cooperation. I wouldn't say they have "several common enemies". While both SK and Japan may have some disputes with China, they are no more military enemies with China than the US is, though all are certainly wary. With respect to Russia, SK has decent relations and while Japan and Russia have some disputes I would not call them enemies.
Outside of the current DPRK situation I wouldn't expect much cooperation between the two, as even though they share a common ally in the US and a common looming regional power in China, the deep cuts that Japan inflicted on its neighbor during WWII have yet to fully heal, with the topic of comfort women being a particularly sour point. There's a lot to delve into on those relations, but I think that falls outside this question.