Question
I have some mature norway maples. I have noticed some borer emergent holes. A tree service told me I have apple tree maggots. How can I be sure and how can I save the existing trees this fall and next spring

Answer
I know one thing it is NOT apple tree maggot. these insects infest the fruit of apples and other fruit trees.

He may have meant Flatheaded apple-tree borer. There are several that may attack Maple.
Shade tree borers are insects that develop underneath the bark of trees and shrubs. Certain beetles and moths are the most common borers.
Most shade tree borers can successfully attack only trees that are injured or stressed.
Shade tree borer development takes from one to three years to complete. Adult stages of these insects occur outside the tree.
Control of shade tree borers includes maintaining growth and vigor of the tree or shrub.
I would spray the trunk with an insecticide called Merit,
Active ingredient; Imidacloprid. Merit is a new generation of insecticides. It is a hard insecticide, but leaves a light foot  print on the Earth. It is labeled with the CAUTION signal word which means it is also very safe to use.  Merit offers tremendous control over a broad range of pests and or conditions that have little or no effective biological control. We strongly urge  those of you with severe pest infestations that can be treated systemically to try this product.

It can be applied as a spray to the trunk and as a soil drench that the roots pick up. I would do both.
Trees that are attacked by borers usually are not in as good health so I would recommend that you fertilize the tree with 10-10-10 at the rate of 1 lb per inch of trunk diameter. Scatter the fertilize around the tree under the limbs and water in good. fertilize now and again in the spring.