You can create the season date ranges and name them by using the Grouping Exposed filter option.
You can find it in the regular Date filter, select
Grouped filters instead of
Single filter for the
Filter type to expose option.
Set the operators to Is between and enter the dates for your seasons (you can see spring and summer in this screnshot):
Drupal will assume midnight if you ommit the time and only include the date. You can look up the exact astronomical times for the seasons if you like :)
Be careful not to exclude some results if you use the same date in separate grouped options but not the same time.