With GNU date and a shell loop:
until [ $(date +%A -d $(date +%Y-%m-$(printf '%02d' "$day"))) = "Saturday" ]
date +%y%m$(printf '%02d' "$day")
This initializes a "day" variable to 1 to indicate the first day of the month, then loops forward day by day until the name of that day is Saturday. GNU date is required in order to use the
-d flag to use a timestamp different from "now". The test for the loop breaks down like this:
[ $(date +%A ...) = "Saturday" ] -- compare the output of date (asking it for the current locale's full weekday name) to "Saturday"
date ... -d $(date +%Y-%m-$(printf '%02d' "$day")) -- give that date command a date to display (don't use the current time) with
-d; that date is generated from a second, inner date command. That inner date command prints the current time in %Y-%m-(day) format, where %Y is the current year, %m is the current month, and we pass in the loop-variable 'day'. The innermost printf statement simply ensures that the day field is zero-padded to two characters.
At the end, it prints out the current year and month in your desired format, using the calculated Saturday day.