No, these are different languages. However, the requests I receive suggest that quite a few people believe Dutch people can speak Danish just as easily. While it's understandable that some might not realize the Dutch are not native Flemish speakers, one would need a native Flemish speaker for that.
To sum it up:
In the Netherlands, the Dutch live and speak the Dutch language. We call ourselves Nederlanders, living in Nederland, and we speak Nederlands.
In Germany, the Germans live and speak German. They call their country Deutschland and the language Deutsch. That word looks like Dutch, but it's not the same.
In Belgium, people speak three languages, one of which is Flemish, which is actually Dutch. However, it’s not the same Dutch spoken in the Netherlands.
And Denmark has absolutely nothing to do with any of the above.