You mention "early exercise" in your title, but you seem to misunderstand what early exercise really means. Some companies offer stock options that vest over a number of years, but which can be exercised before they are vested. That is early exercise. You have vested stock options, so early exercise is not relevant. (It may or may not be the case that your stock options could have been early exercised before they vested, but regardless, you didn't exercise them, so the point is moot.)
As littleadv said, 83(b) election is for restricted stocks, often from exercising unvested stock options. Your options are already vested, so they won't be restricted stock. So 83(b) election is not relevant for you. A taxable event happen when you exercise. The point of the 83(b) election is that exercising unvested stock options is not a taxable event, so 83(b) election allows you to force it to be a taxable event. But for you, with vested stock options, there is no need to do this.
You mention that you want it not to be taxable upon exercise. But that's what Incentive Stock Options (ISOs) are for. ISOs were designed for the purpose of not being taxable for regular income tax purposes when you exercise (although it is still taxable upon exercise for AMT purposes), and it is only taxed when you sell. However, you have Non-qualified Stock Options. Were you given the option to get ISOs at the beginning? Why did your company give you NQSOs?
I don't know the specifics of your situation, but since you mentioned "early exercise" and 83(b) elections, I have a hypothesis as to what might have happened. For people who early-exercise (for plans that allow early-exercise), there is a slight advantage to having NQSOs compared to ISOs. This is because if you early exercise immediately upon grant and do 83(b) election, you pay no taxes upon exercise (because the difference between strike price and FMV is 0), and there are no taxes upon vesting (for regular or AMT), and if you hold it for at least 1 year, upon sale it will be long-term capital gains. On the other hand, for ISOs, it's the same except that for long-term capital gains, you have to hold it 2 years after grant and 1 year after exercise, so the period for long-term capital gains is longer.
So companies that allow early exercise will often offer employees either NQSOs or ISOs, where you would choose NQSO if you intend to early-exercise, or ISO otherwise. If (hypothetically) that's what happened, then you chose wrong because you got NQSOs and didn't early exercise.