This should do the trick:

WITH cteEmployee As
(
  SELECT
      Id,
      DeptId,
      quality,
      VerifiedOn,
      ROW_NUMBER() OVER 
      (
          PARTITION BY 
              DeptId 
          ORDER BY 
              CASE quality WHEN 'Best' THEN 0 ELSE 1 END, 
              VerifiedOn DESC
      ) As RN
  FROM
      Employee
)
SELECT
  D.Id,
  D.Name,
  E.Id,
  E.quality,
  E.VerifiedOn
FROM
    Department As D
    LEFT JOIN cteEmployee As E ON E.DeptId = D.Id And E.RN = 1
ORDER BY
  D.Name
;

Results:

ROW_NUMBER (Transact-SQL) – SQL Server | Microsoft Learn

I tested the query on a DB Fiddle on MySQL v9 and I got your expected results.

What this does: I use ROW_NUMBER() and PARTITION BY e.DeptID to rank the employees in each department.

Then I sorted them first by Quality and secondly by VerifiedOn

This allows me with rn = 1 to return only the latest “Good Quality” employee from every department

On the Final SELECT its just a LEFT JOIN between both.

To ensure departments with no employee who meets the requirements COALESCE(le.EmpID,0) replaces any NULL values with 0

  WITH LatestEmployee AS (
        SELECT 
            e.Id AS EmpId,
            e.DeptId,
            e.Quality,
            e.VerifiedOn,
            ROW_NUMBER() OVER (
                PARTITION BY e.DeptId
                ORDER BY 
                    CASE WHEN e.Quality = 'Best' THEN 1 ELSE 2 END,
                    e.VerifiedOn DESC
            ) AS rn
        FROM 
            Employee e
    )
    
    SELECT 
        d.Id AS DeptId,
        d.Name AS DeptName,
        COALESCE(le.EmpId, 0) AS EmpId
    FROM 
        Department d
    LEFT JOIN 
        LatestEmployee le ON d.Id = le.DeptId AND le.rn = 1
    ORDER BY 
        d.Id;

I used 3 sub queries to search for the employees.

SELECT d.*, 
  COALESCE
  (
    (SELECT top 1 e.ID 
     FROM Employee e 
     WHERE e.deptID=d.Id AND e.quality='Best' 
     ORDER BY VerifiedOn desc),
    (SELECT top 1 e.ID 
     FROM Employee e 
     WHERE e.deptID=d.Id AND e.quality='Good' 
     ORDER BY VerifiedOn desc),
    (SELECT top 1 e.ID 
     FROM Employee e 
     WHERE e.deptID=d.Id AND e.quality='Ok' 
     ORDER BY VerifiedOn desc), 
    '0'
  ) as EmpiId
FROM Department d
GROUP BY d.ID,d.Name

Alternative suggested by Siggemannen:

SELECT d.*, 
  ISNULL  
  ( 
   (SELECT top 1 e.ID 
   FROM Employee e 
   WHERE e.deptID=d.Id 
   ORDER BY CASE  e.quality
            WHEN 'Best' THEN 0
            WHEN 'Good' THEN 1
            WHEN 'OK' THEN 2
            Else 3
            END,
            VerifiedOn desc),
   0
  )  as EmpiId
FROM Department d
GROUP BY d.ID,d.Name

fiddle