Yes indeed, SpaceX has announced publicly about uncrewed landings on the Moon before 2022 !
From the article Spacex wants to land Starship on the Moon before 2022, then do cargo runs for 2024 human landing:
"Aspirationally, we want to get Starship to orbit within a year", Shotwell said.
"We definitely want to land it on the Moon before 2022. We want to [...] stage cargo there to make sure that there are resources for the folks that ultimately land on the Moon by 2024, if things go well, so that's the aspirational time frame. "
That's an ambitious timeline, and as Shotwell herself repeatedly stated, these are "aspirational" timelines. In the space industry, aswell as in tech, it's not uncommon for leadership to set aggressive schedules in order to drive the teams working on projects to work at the limits of what's actually possible.
(emphases by me)