Ohhhh, well yup! It's mentioned in the conversation between Anakin and Padme on Naboo. In Attack of the Clones, Padme was the target of a few assassination attempts in, and on the way to, Coruscant. So to protect her, the Jedi Order sent her back to Naboo with Anakin as a bodyguard, while Obi-Wan searched for the assassin.
When the pair arrived on Naboo, they went to visit the Queen to explain the situation. On the way, there is a scene where Padme and Anakin are walking together, with R2D2. And Padme did mention that she felt she had been too young when the people of Naboo asked her to serve as Queen, and she felt that she had not been ready at the time. But after her 2 terms were over, the new Queen asked her to stay on and continue to serve Naboo as a senator in the Senate. And she just couldn't refuse.
So, yup. Padme was the Queen when Anakin met her. Then she served out her 2 terms as Queen, probably while Anakin was off training as Obi-Wan's apprentice (while he was away and didn't see her again for 10 years). After her 2 terms were up, she simply continued to help the next Queen, by serving in the Senate.
Also, in the movies, the Queen of Naboo that Anakin and Padme met in the second movie, and the Queen that walked in Padme's funeral procession in the third movie, are also different. The Queen in the second movie is obviously older, while the Queen in the third movie looked a lot younger (though we only caught a glimpse of her). So the second movie's Queen had probably also served out her 2 terms, and had given up the seat after that to the new Queen. Looks like she didn't serve as Senator after that, though...