The output of today() gives the date in this format: 2025-01-25
For getting 25 Jan 2025, it should be possible to write a space-script function to convert the output of today() into this format.
After that you could call that space-script function inside the QuickNote template and it should display date like 25 Jan 2025 in all daily quick note pages.
Or if you wanted flexible control over the date formatting, you should be able to design the space-script function accordingly. The input to the space-script function can be the date format specification like you shared, and the function could apply that formatting to the output of today().
Maybe this discussion could give some hints: Format today function to YYYYMMDDHHmmss
Is this what you were looking for?