Corrected sql is-
declare @Qtr int =1;
update Appraisal_QtrUsers_Details
set Q1AppseKRA_SubDate=(case when @Qtr=1 then null else Q1AppseKRA_SubDate end),
Q2AppseKRA_SubDate=(case when @Qtr=2 then null else Q2AppseKRA_SubDate end),
Q3AppseKRA_SubDate=(case when @Qtr=3 then null else Q3AppseKRA_SubDate end),
Q4AppseKRA_SubDate=(case when @Qtr=4 then null else Q4AppseKRA_SubDate end)
where user_id=9177
The logic behind this query is if the column is not supposed to be updated with NULL value then just update it with the its current value and will not result in any value change.
Note: It would be better if you can share some business logic for this query so that it would be easier to suggest you more correct solution.
Thanks :)