If I flied a claim with the city due to a tree branch that fell on my car but the claim is denied, can I hire an attorney to appeal/dispute the claim denial?

Yes, you can hire an attorney to dispute the denial, but the real question is whether it is worthwhile to do so. The city would only be liable for the damage done by the branch if they had reason to know that this branch or tree posed some hazard or risk and, despite knowing of that enhanced or specific risk, failed to take action. Or alternately, you'd have to show that the city was generally negligent, or unreasonably careless, in how it took care of its trees. That would mean you'd have to be able to show that someone had contacted the city to warn them about this tree or branch before the incident or that the city was unreasonably far behind in generally inspecting or taking care of trees. Neither showing is likely to be easy, and may not be possible but if you can't show show one of these things, you won't be able to show that the city was at fault and if the city was not at fault, they would not be liable. So you could spend alot of money on a case you lose and even if you could win, if you hire a lawyer, you will most likely spend several thousand dollars on the case, which may be more than you would recover if you win. You generally can't get your legal costs from the other side, but have to pay them yourself.