You have the first single quote in the wrong place, it should be after the =.
sqlString = "UPDATE tbl_RETRO_EFF_76_PREV_WK AS t1 INNERJOIN tablec AS t2 ON t1.HIC_NUMBER = t2.HIC_NUMBER set"
sqlString = sqlString & " [COMPLETED] =" & c.Value & ","
sqlString = sqlString & " [DATE] = #" & c.Offset(0, -1).Value & "#,"
sqlString = sqlString & " [NTID] ='" & c.Offset(0, -2).Value & "';"
By the way, a good way to check the SQL statement is to print it out to the debug window (CTRL+G) with ?sqlString, copying it and then pasting it into the SQL view of a query in the database