Monthly grouped data
You can use your CASE variant with some modifications. First you need to extend the GROUP BY with MONTH(date): 

SELECT product_name,
  CASE WHEN MONTH(date) = 1 THEN SUM(quant_ton) END AS JAN,
  CASE WHEN MONTH(date) = 2 THEN SUM(quant_ton) END AS FEV
FROM tb_import
JOIN tb_cost
    ON tb_import.productid = tb_cost.productid
JOIN tb_sites
    ON tb_sites.site_num = tb_import.site_cd
JOIN tb_product
    ON tb_product.prod_id = tb_import.productid
GROUP BY product_name, MONTH(date)


If not MySQL will randomly pick one month for you. There's a lot to be said about this default setting (insane comes to mind:-), but look up sql_mode ONLY_FULL_GROUP_BY for ways around it.

As you can see now you will get null for non-matching months:

product_name    JAN    FEV
gravel          41     (null)
gravel          (null) 22
soil            23     (null)
soil            (null) 10


By picking - for example - max of JAN and FEV you will get rid of those

SELECT product_name, max(JAN), max(FEV)
FROM (
    SELECT product_name,
        CASE WHEN MONTH(date) = 1 THEN SUM(quant_ton) END AS JAN,
        CASE WHEN MONTH(date) = 2 THEN SUM(quant_ton) END AS FEV
    FROM tb_import
    JOIN tb_cost
        ON tb_import.productid = tb_cost.productid
    JOIN tb_sites
        ON tb_sites.site_num = tb_import.site_cd
    JOIN tb_product
        ON tb_product.prod_id = tb_import.productid
    GROUP BY product_name, MONTH(date)
) as t
GROUP BY product_name;

product_name    max(JAN)    max(FEV)
gravel          41          22
soil            23          10


However, the pivoting is only a presentation of the result. I would choose a query like:

SELECT product_name,
       MONTH(date),
       SUM(quant_ton) 
FROM tb_import
JOIN tb_cost
    ON tb_import.productid = tb_cost.productid
JOIN tb_sites
    ON tb_sites.site_num = tb_import.site_cd
JOIN tb_product
    ON tb_product.prod_id = tb_import.productid
GROUP BY product_name, MONTH(date)
ORDER BY product_name, MONTH(date);


and then format the result in the presentation layer of the application.