Whether or not your HVAC is covered by your homeowner's policy depends on what is wrong. If the unit was damaged or destroyed by one of the named perils in your home insurance policy, it is probably going to be covered. If it simply stopped working for apparent reason, then you are probably going to have to pay for the repairs out of pocket. Insurance is not meant to pay for the repair or replacement of personal property for any reason, only for reasons covered by the policy. You can still try to submit a claim, but don't be surprised if it gets denied.
There is a huge difference between a damaged HVAC and one that is nonfunctional, in the eyes of the insurance company. Typically, the working status of your household appliances is covered under a home warranty, not by your homeowners insurance. Home warranties are special kind of insurance that guarantees your appliances for the period of time listed in the policy, while home insurance is limited to protecting you against lost or failed appliances in the event of a covered peril. In the first case, appliances will break down and need to be repaired, while the second case is reserved for things such as fire, wind storms and other perils named specifically in your homeowners policy.
One example would be where the compressor on your HVAC unit goes out. While this could lead to some uncomfortable evenings, and it does lower the value of the home, the HVAC unit is not considered a necessary appliance for being able to live in the home. If the HVAC is not covered under a regular warranty and you did not purchase a home warranty when you bought the home, then you will need to find and pay a contractor out of pocket to get the compressor replaced or repaired.
If you believe that the HVAC unit should be covered under your home insurance, contact your insurance agent and find out. If it is covered, the insurance company can tell you how to proceed. If it is not covered, the insurance will tell you so. The same information is available in your policy, so be sure to start there, looking for specific exclusions to home appliances.