You can use Default Choices and use today's date in there (as one of the preset piped text fields).
Alternatively you can use one of the inbuilt Qualtrics questions to have a calendar type question, and from memory it defaults to today's date. I know I've done it previously but it was a few years ago now and can't remember if that was a setting to make it default to today (using default choices mentioned above), or whether it was automatically the default!
Both of the above options will be changing the question from your current drop down setup. Third option allows you to keep your current drop down but you could use Embedded Data in your Survey Flow to capture today's date, and then in your default choices options, you could set the default choice to be that embedded data field. Provided they both have the same date format, it should work the way you are looking at.
Hope these options help!