I received $16,500 from a family trust when my grandmother passed away in 2009. My mother (the person in charge of trust) informed me the family accountant said I would not be receiving a 1041/K-1 and I would not need to pay taxes on this inheritance because the trust was already paying taxes on the money. Is this true?

Do I need to claim this inheritance somewhere and pay taxes on it, or am I in the clear?

You're in the clear if the accountant says you are. The only time you have to pay income tax from a trust like this is if they issue you a schedule K-1, and in that case you are paying tax on the income the trust receives between the date of death and the distribution of the principal to the heirs, not on the value of the inheritance itself. So if the accountant isn't sending out a K-1, you're in the clear. There is no federal inheritance tax, so no wories there. The only possible concern is if your state has an inheritance tax (a few do, so you should investigate that).

So I don't even have to claim it anywhere on my taxes? I don't pay state tax because I'm in Florida, and I'm active duty military.

So I don't even have to claim it anywhere on my taxes? I don't pay state tax because I'm in Florida, and I'm active duty military.

Correct - you can ignore it for tax reporting purposes.