When one claims that one's argument is correct because soley of one's driving achievements, then one is appealing to one's accomplishment. This is a type of genetic fallacy.
Let's rewrite your question as a syllogism:
P1. Driving the car in a particular manner causes accidents.
P2. I have driven in such a way I have never been in an accident.
C. Therefore, I'm an expert on driving and avoiding accidents.
The conclusion is what the argument purports to be about. In this case, the claimant is making an argument to prove they are an expert in driving.
One potential fallacy at play would be the genetic fallacy:
The genetic fallacy (also known as the fallacy of origins or fallacy of virtue)1 is a fallacy of irrelevance that is based solely on someone's or something's history, origin, or source rather than its current meaning or context.
The question is 'does this sole premise to the claimant's driving history serve as an adequate basis for determining the driver's current expertise?' Obviously not. Let's us imagine that a famous racecar driver in the 1930's was alive, and talks about how he won races and was an excellent driver, but now he is in his 90's suffers from legal blindness, and hasn't driven in 20 years. Does his history allow us to conclude with certainty that he is currently an expert? Absolutely not. But we can get more specific. In this argument, the claimant appeals to his accomplishments to validate his claim to expertise.
Another avenue of attack on this argument is that P2 purports to be an accomplishment, and certainly if this same driver had not only a racing pedigree, but also a flawless domestic record in driving, that would be quite the accomplishment; however, is this driver still an expert on driving given his circumstances? Not necessarily, so here we see that the appeal is to accomplishment which is a type of genetic fallacy. From the WP article:
Description: When the argument being made is sheltered from criticism based on the level of accomplishment of the one making the argument. A form of this fallacy also occurs when arguments are evaluated on the accomplishments, or success, of the person making the argument, rather than on the merits of the argument itself.
Remember, a general notion of expertise in driving would require not only current knowledge-that but knowledge-how. The question of a driver's expertise isn't a question of prior, irrelevant accomplishments, but rather the current state of the driver's ability. So, while invoking accomplishments in a larger framework of argument certainly strengthens an argument, relying on it in the absence of other factors would be specious.
Please note that though the conclusion that the argument arrives at, that the driver is an expert, is false, this is NOT an an appeal to false authority as some have suggested. From the article:
A similar fallacy is the appeal to false authority. This fallacy is used when a person uses a false authority as evidence for their claim.
Such an argument relies on a premise invoking authority and not in the conclusion. The question at hand is 'Is the conclusion of authority invalid?', not 'Is the claim based on invalid authority?'.
Therefore, the best fit for this fallacious reasoning would be the appeal to accomplishment.