Hi, Anne -- Thanks for reaching out to the UNH Extension Yard and Garden Infoline regarding your question about hemlock tree needles turning brown in spots.

Over the past seasons, the most commonly reported problem with hemlocks has been  an invasive aphid-like insect called the hemlock wooly adelgid or HWA. These insects feed on hemlock trees and are identified by an abundance of small fuzzy white masses attached to the younger branches of a hemlock tree. If the wooly adelgid is causing the brown needles, you should be able to see the insects on the branches. The adult HWA reveals its presence most of the year by a dry, white woolly substance it secretes after it settles on young hemlock twigs. Identifying wooly adelgid

If left alone, these insects can weaken and eventually kill hemlock trees. In times of extended drought and thus higher stress levels on trees, damage from HWA can be more severe and can lead to faster die-off. It's important to identify an infestation early in order to successfully treat it and avoid spreading it to uninfected trees. For more information about the hemlock wooly adelgid and recommendations for control, please see the resources listed below- 

Wooly Adelgid Fact Sheet

Controlling Wooly Adelgid

Please note, however, that hemlocks are prone to other diseases as well.  If you have inspected the branches and do not see the wooly adelgid, you might want to consider other possibilities as shown in this link. Hemlock Diseases

Also, for a list of certified arborists in New Hampshire, see the New Hampshire Arborists Association website here.

I hope this helps and feel free to reach out with any further questions!