Can I use both flying kick and Dragon Style in the same round?

Yes, when you have Dragon Ferocity, while using Dragon Style, all of your unarmed strikes benefit, regardless of when or how they are used—regular attack, full attack, flurry of blows, charge, attack of opportunity, whatever. Dragon Ferocity just improves your unarmed strikes, however they happen.

As a style strike, flying kick can be used during a flurry of blows, and as it is a kick, it is an unarmed strike. It therefore benefits from any improvements you may have to unarmed strikes, such as from Dragon Ferocity.