Probably. Starship according to nasa won't launch until December at the earliest. So that means January at the earliest. They have to test out a lot of things in earth orbit after the first test. Mostly cryogenic fluid transfer and re-entry testing. Currently they have multiple ships and boosters in the works. So I imagine all of next year will be orbital test flights every couple of months to test out necessary functions for landing humans on the moon in 2025 for nasa.

In 2024 they'll are supposed to do the uncrewed demonstration to the moon. I doubt they will want to send starships to mars near then so as not to make nasa feel like they aren't focused on landing on the moon. I know in the past when working on future projects some at nasa felt they may be distracted from commercial crew.

If they land humans on the moon in 2025 then I could see demonstration missions to mars in the window in 2026. Mostly because they proved starship is safe at that point so any mishaps landing on Mars won't have a big effect on the moon missions. But when it comes to mars that means if they don't do the moon landing until 2026 they'll have to wait til 2028 for the next mars window. That's the earliest I expect a cargo/demonstration mission.