The short answer is usually no, and it is better to hear that plainly than to spend weeks finding out.
Section 767.04 makes the owner of the dog liable to the person bitten. Where those are the same person, there is nobody to claim against. Homeowners and renters policies reinforce it: liability cover is for injuries to other people, and injuries to the policyholder and members of the household are excluded almost universally.
A few narrower situations are worth checking before you accept that:
The dog was not legally yours. A dog you were fostering, minding, walking, or had not yet formally taken on may still belong to someone else — and the statute follows ownership.
Someone else caused it. If a third party provoked, struck, or released the dog, or if a trainer or boarding facility handled it in a way that caused the injury, the claim runs against them rather than through the dog.
You were bitten at work. A dog you handle professionally — kennel, veterinary practice, grooming, rescue — is a workers’ compensation question, which is a different system with a different answer. More on bites at work.
Your own health cover. Not a claim, but worth confirming what your medical policy pays, and getting the wound treated properly. Dog bite infection is the complication that turns a minor injury into a serious one, and it does not care whose dog it was.