Sort of like this:

create table [dbo].[mytable]
(
[StartDate] date not null primary key
, [CalendarMonth] as datepart(month,[StartDate]) persisted
, [CalendarYear] ...

Also, you really shouldn't be using a loop for this.  You could put this into the table from your other thread.

It probably needs some additional work but I will leave that to you to at least attempt first.  Questions or problems please come back here.

Here is a partial solution.  It doesn't incorporate the WHERE clause for the time sheet information.

/****************************************************************************************
Generate employee data
****************************************************************************************/
if object_id('[dbo].[Employee]','U') is not null
...

Use the table in your other thread here, https://www.sqlservercentral.com/Forums/2008707/get-qtrmonth-and-year-from-start-date?Update=1#bm2008841, using the [fy_year] and [fy_month] columns to create the YYYY-MM, like this:
CAST([fy_month] as varchar(4)) + '-' + right('0' + cast([fy_month] as...

Just one thing missing, to get the expected results how is the query run.  I am trying to figure out how the employees come together using the start and end...

Okay, here you go.

create table [dbo].[mytable]
(
  [StartDate]   date  not null primary key
, [CalendarMonth]  as datepart(month,[StartDate]) persisted
, [CalendarYear]  as datepart(year,[StartDate])...