It could be possible with enough training of the dog to train the blue heeler into becoming a hunting dog but it would take a lot of training and hard work to get the Blue Heeler to be a hunting dog.
Blue Heeler dogs are mostly known as work dogs where the Blue Heeler herds cattle, sheep, horses and other livestock on farms and ranches.
A good hunting dog that works great for retrieving the birds and other game that you shot would be a Golden Retriever.
Golden Retrievers are really great hunting dogs and that's where the last name of the Golden Retrievers name "Retriever" comes from.