You only need to add the following line:
COUNT(gs.SerialNumber0) AS NumOccurences
Whole code:
SELECT
cs.Name0 AS [Computer Name],
rs.User_Name0 AS [User Name],
rs.User_Domain0 AS [Domain],
gs.SerialNumber0 AS [Serial Number] ,
COUNT(gs.SerialNumber0) AS NumOccurences
FROM v_GS_COMPUTER_SYSTEM cs JOIN v_R_System rs ON cs.resourceID = rs.resourceID
JOIN v_GS_PC_BIOS gs ON cs.ResourceID = gs.ResourceID
GROUP BY
cs.Name0,
rs.User_Name0,
rs.User_Domain0,
gs.SerialNumber0
HAVING COUNT (cs.Name0) > 1