Comment 1 for bug 1334201

> When importing the products from Magento, there has been no stock update in Odoo

This is normal. You should create in initial inventory.

> when having stock in Odoo there is no update in Magento, even when running "Update all stock" in the backends configuration.

- Are export jobs created?

Can you try to:
 1. edit a product
 2. in the 'connector' tab, open the binding, see the stock level
 3. change the quantity of the product
 4. open the binding again, click on 'update' next to the stock level: it should be updated
 5. check if a job has been created