If we shifted the calender by 10 days, could we get the 4 quarters to roughly fit with the start of....?

I'm confused! If you are talking about the seasons, those dates are related to the solstices...longest day, shortest day of the year. You cant just "slide things around" for convenience. There are reasons the start of seasons are on the days they are (and why they sometimes shift 20th or 21st).