Generally dates expressed as strings in database queries should be in the format "YYYY-MM-DD".
You essentially want your executed query to be this:
SELECT * from mating
WHERE date BETWEEN '2013-12-20' AND '2014-02-20'
So change the format of your dtpickers to be yyyy-mm-dd like this:
export_daily_date_DateTimePicker1.Format = DateTimePickerFormat.Custom
export_daily_date_DateTimePicker1.CustomFormat = "yyyy-MM-dd"
export_daily_date_DateTimePicker2.Format = DateTimePickerFormat.Custom
export_daily_date_DateTimePicker2.CustomFormat = "yyyy-MM-dd"