Just simple as bellow:

DELETE FROM clients 
WHERE id NOT IN (
    SELECT * FROM (
        SELECT c.id 
        FROM products p
        INNER JOIN clients c 
            ON c.id = p.supplier_id
        WHERE c.name = 'Q-IMAGING AUSTRALIA LTD (HEAD OFFICE)'
        GROUP BY c.id
    ) client_tbl
)
AND name = 'Q-IMAGING AUSTRALIA LTD (HEAD OFFICE)' 
AND type = 'Supplier'