Is the frog used to represent a year in the Chinese cycle of years?

No, The frog never represent a year in Chinese Cycle of Years.

Here's why: The Chinese Cycle of Years can be classified to three categories: 1: The six animals that are tamed by ancient Chinese: Ox, Sheep,Horses, Pigs, Chicken, and Dogs. 2: Animals that are close to everyday life: Tigers, Rabbits, Monkeys, Rats, and Snakes. And the last category, the animals that symbolized China -- Dragon, which is this year. Dragon symbolized prosperity and peace, so the Chinese Cycle of Years must have Dragon.

Well, since you love frogs so much, I will tell you a good news: Frog is symbolized as full moon in China. And in Chinese folk tales, the frog loves the smell of golden coins, and it will bring back them to the owner. So if you are a business man, put a statue of a frog (toad is better) with a coin in its mouth, you will be success in business.