If the fire dept in their endeavor to put out a house fire.
Inadvertently causes damage to an adjacent property.
Could the offended property owner seek relief for damages?
Or is the fire dept immune from paying damages?

Could the offended property owner seek relief for damages? YES
Or is the fire dept immune from paying damages? NO

Some municipal legal departments are terrible, however, and will flat out tell you to sue them, because they aren't going to handle a claim in any sort of honorable way. That seems to be the way it is in cities more than small towns.
Take lots of dated pictures and document everything well, including receipts for repairs, or quotes from contractors.
Ask first if the city or town wants to deal with a claim before you sue, but get their answer in writing, and don't wait too long before hiring a lawyer. Get one on contingency, if you can.
If it's a few thousand dollars, try small claims first.
Collecting even if you win is another matter, and if they don't pay, you have to go back to court at least once or twice again.
if it's more minor like a broken fence and some lawn damage, try shmoozing the Fire Dept. They probably have people who can fix things.

There's also HO insurance - both homeowners, starting with the one who was damaged, who might handle it with the insurer of the other home.

The key here is negligence. If it can be proven that there was negligence in protecting the adjacent property, then yes a case may be won. But that may not be easy to prove.

Thank you

Laws in some locations require that a claimant first give a specified notice to the municipality which operates the fire department, before you can sue. So be sure your attorney checks for any such requirement.

And, as has been suggested, you might file a claim with you homeowner's insurance carrier, and then let them go against the fire department.

Very good advice

It is also possible, that because of the event, the homeowners insurance of the house on fire, may have some responsibility because the fire department was there and the ire was the causing event.

The fire department can be held liable, along with the actual person, (fireman) that caused the damage if it was because of carelessness or not following procedure properly.

I think it depends on the type of damage done. Was it due to negligence on behalf of the fire department, or was it due to putting out the fire at your neighbors house? In other words, was it water damage due to the hoses used to stop the fire at your neighbors house, which sadly is next to your home? Was there fire damage because the fire spread, etc? Or did the fire truck drive over your lawn, knock down a fence, or do something for which it's actually negligent?

Yes it was from the fire cannon. Nothing major. The house has been abandon
For 9 years. It is still in foreclosure proceedings.

Which house was abandoned, the one on fire or the neighboring one? Depending on the proximity of the house, water damage from the hose spray may have been unavoidable.