SET @x := '2018-06-01';

SELECT CASE WHEN WEEKDAY(@x) = 4 
            THEN CASE WHEN DAYOFMONTH(@x) < 8 
                      THEN 'FIRST FRIDAY' 
                      ELSE 'SOME OTHER FRIDAY' END 
            ELSE 'NOT FRIDAY' END x;
+--------------+
| x            |
+--------------+
| FIRST FRIDAY |
+--------------+

SET @x := '2018-06-08';

SELECT CASE WHEN WEEKDAY(@x) = 4 
            THEN CASE WHEN DAYOFMONTH(@x) < 8 
                      THEN 'FIRST FRIDAY' 
                      ELSE 'SOME OTHER FRIDAY' END 
            ELSE 'NOT FRIDAY' END x;
+-------------------+
| x                 |
+-------------------+
| SOME OTHER FRIDAY |
+-------------------+

SET @x := '2018-06-07';

SELECT CASE WHEN WEEKDAY(@x) = 4 
            THEN CASE WHEN DAYOFMONTH(@x) < 8 
                 THEN 'FIRST FRIDAY' 
                 ELSE 'SOME OTHER FRIDAY' END 
            ELSE 'NOT FRIDAY' END x;
+------------+
| x          |
+------------+
| NOT FRIDAY |
+------------+