find count of Saturday and Sunday count in a year with out loop,cte

You can try the following query:

DECLARE @startdate DATETIME = '2022-01-01'
DECLARE @enddate DATETIME = '2022-01-31'
DECLARE @TotalSundays INT
DECLARE @TotalOtherDays INT

SET @TotalSundays = (SELECT  DATEDIFF(DAY,@startdate,@enddate) /7 +
CASE   
    WHEN   DATEPART(WEEKDAY,@StartDate) = 1 THEN 1
    WHEN DATEPART(WEEKDAY,@StartDate) > DATEPART(WEEKDAY,@EndDate) THEN 1
    ELSE 0
END)

SELECT @TotalSundays [Total Sundays]

This query helps to you
 
Declare @StartDate as datetime = '01/Jan/2022'
Declare @EndDate as datetime = Getdate()

select count(*) as Daycount
from master..spt_values as Number
where Number.type = 'P' and
dateadd(day, Number.number, @StartDate) <= @EndDate and
datepart(dw, dateadd(day, Number.number, @StartDate)) = 5