As far as I understand, the problem with that is Google Shopping requires an e-commerce store to buy from. In effect, it's just a link back to your product in your store.
As linnworks isn't an E-commerce platform you can't really use it to populate a Google Shopping feed as it requires more data than it has access to. You would need to create a XML feed for google shopping from your store. There are plugins for all the big e-comm platforms that will do the job.
Google Shopping (Google Express) does allow you to mark some items in your product feed as Google Express items. These items are then treated differently and are listed on the Google Express platform. Google Express does generate the order and process the money for the transaction, similar to Walmart, Amazon, eBay... So really this is another selling platform that Linnworks should integrate with ASAP. Currently we are having to collect orders and manually import into Linnworks, so it would be great if you built out the integration. Would be a great offer to attract new customers. I am sure they want a piece of the pie like Amazon and Walmart.
Example of Google Express listing
Please add Google Shopping Integration to Linnworks!!! Its a no brainer