Were there any battles near New Guinea in World War 2?

Definitely.
In January 1942, the Japanese invaded New Guinea with the intent of controlling the island, hoping to increase the extent of Japanese control in the Pacific. They held several bases on the north coast, as well as Rabaul.

The Japanese tried to make their way over the Owen Stanley Range to port Moresby. This was where the battle of the Kokoda Trail was fought, as Australian and US troops, aided by PNG natives, repelled the Japanese troops. A year after their invasion, the Japanese were forced to withdraw from Papua New Guinea.
Definitely.
In January 1942, the Japanese invaded New Guinea with the intent of controlling the island, hoping to increase the extent of Japanese control in the Pacific. They held several bases on the north coast, as well as Rabaul.

The Japanese tried to make their way over the Owen Stanley Range to port Moresby. This was where the battle of the Kokoda Trail was fought, as Australian and US troops, aided by PNG natives, repelled the Japanese troops. A year after their invasion, the Japanese were forced to withdraw from Papua New Guinea.