My husband and I are looking for an apartment in NH and the landlord we just saw said because of our 4 pound medical support dog... we would need first months rent and security deposit plus another whole security deposit for our dog... is this legal?

It depends whether your dog is classified as a "service animal." If it is, then under the Fair Housing Act:

"Under the reasonable accommodation provisions of The Fair Housing Act, providers are required, upon request, to make an exception to a “no pets” rule and cannot apply pet-related restrictions or conditions on a service animal. This means that restrictions on breed, size, weight and the number of animals per household cannot be applied to service animals. Likewise, service animals are allowed in all public and common areas such as dining facilities, laundry rooms, swimming pools and clubhouses. No liability insurance or veterinary certificates can be required and no extra fees or additional security deposits can be charged. [emphasis is mine] However, if a service animal causes damage to the unit or the common areas of the dwelling, the tenant may be asked to pay for damages if it is the housing provider’s policy to ask for compensation from non-disabled residents."

See The Seeing Eye - Learn About Guide Dogs in Housing for additional info.

So the question is whether your dog is a service animal or not. From the ADA web site https://www.ada.gov/service_animals_2010.htm

"Service animals are defined as dogs that are individually trained to do work or perform tasks for people with disabilities. Examples of such work or tasks include guiding people who are blind, alerting people who are deaf, pulling a wheelchair, alerting and protecting a person who is having a seizure, reminding a person with mental illness to take prescribed medications, calming a person with Post Traumatic Stress Disorder (PTSD) during an anxiety attack, or performing other duties. Service animals are working animals, not pets. The work or task a dog has been trained to provide must be directly related to the person’s disability. Dogs whose sole function is to provide comfort or emotional support do not qualify as service animals [emphasis mine] under the ADA."

This is what federal law says. I don't know whether NH offers any additional protections regarding "medical support dogs."

States are cracking down on service animals. Online sites are selling certificates and dog coats to anyone. You may have to have documented evidence that your dog meets the definition.

I doubt that 'medical support' meets the definition, or that a 4 lb dog can do much. For many, that means emotional support, rather than pressing 911 or a life alert.
While I love dogs, I do see a lot of abuse on airplanes and in public buildings.

Landlords want first, last, and security anyway.
Beg for a low dog deposit. Mainly this is for floor damage, and/or lack of clean up around the building, or complaints of barking.

NH law doesn't appear to be specific about pet deposits. Renting, Security Deposits, and Evictions | Consumer Sourcebook | Consumer Protection and Antitrust Bureau | NH Department of Justice

They do specify that any amount not considered part of the monthly rental is a security deposit. And that a security deposit cannot exceed one month's rent. So it might not be legal to charge an extra fee for a pet deposit. The site I linked to has an area where you can try to get the info from NH.