No. The HEV does not have a lead-acid 12V battery, so no ability to trickle-charge. The 12V comes from a small "reserve" of the higher voltage traction battery and is simply converted down to 12V. This is why you get the 12V boost button on the left side of your dashboard. If it ever runs down, you can simply boost the starter motor with it and get the car running again. You can find this process in the manual.
However, the hybrid system is very protective of its traction battery juice (the car is dead without it), so if you run the 12V down, for example by using accessory mode w/o the engine running or leaving the lights on, etc. and also use up too much juice trying to boost it again, it will simply shut down the 12V side.
That's why there is a positive jump-start post in the engine bay right next to the fuse box on the right side. You can attach cables to it and a metal non-moving part of the chassis in there somewhere for the negative to jump-start it, if the boost button ever fails you.
If you aren't using it much, make sure you lock it, as that will shut down some of the 12V equipment in the car.
And lastly, if you are leaving it in storage for weeks or months, there is also a 12V disconnect cable in the engine bay, but on the left side back next to the strut tower. Its the black wire & connector next to the orange high voltage cable. But if you disconnect that, you will also lose all your saved programming - radio stations, seat memory, etc. But better than a dead car when you come back from vacation!