Deer running out in front of you is certainly something to be concerned about. The good news is that hitting a deer is not an at-fault accident. In the eyes of your insurance company, hitting a deer is an unavoidable accident.
In this case, your car insurance is unlikely to increase, even if you file a claim to repair damages to your car. However, it’s still a good idea to review your policy and make sure your insurance company didn’t increase your rates as a result of the claim.
Moreover, you should get an estimate on the damage to see if you need to file a claim or pay it out of pocket. If the damage is less than your deductible, you’re better off just paying for it without filing an insurance claim.