"Ik spreek Nederlands."
Translation:I speak Dutch.
Short answer: it's not wrong, but it's linked to some complicated history and no clear distinction between the languages Dutch and German and their predecessors.
I'm no expert, but what I could quickly find about it:
It goes back to the late Middle ages when a early forms of Dutch and German were spoken: dietsc and duutsc refer to variants of the Germanic languages. These terms were mostly used to distinguish both Dutch and German from Roman languages and can apply to either Dutch or German. The distinction between Dutch and (low-)German (Nederlands en (Neder-)Duits) was not that clear.
The English word Dutch is derived from duutsc or duutsch
Also in Dutch the word Duits (current meaning: German) was used to refer to Dutch (Nederlands) up to the 17th century. This can for instance be seen in the Dutch national anthem (sung from the perspective of Willem van Oranje), the second line is:
ben ik, van Duitsen bloed (am I, of Dutch/German blood)
Also see this explanation in Dutch: https://nl.wikipedia.org/wiki/Nederlands_(naamsgeschiedenis)#.C3.9Eeudisk_in_de_volkstaal
All I'd really add to that would be that as I understand it the term(s) which ultimately became Dutch in English and Deutsch in German meant something like "vernacular" and/or "of our people". The Dutch used the words mentioned to designate themselves and their language as the Germans do today. We (English speakers) adopted their term as our term for them. However, we already had a term for the region that is now Germany (what was the the Holy Roman Empire), which was derived from the Latin Germania, so we used that instead to refer to Germany, German and Germans.
It's as OscarHermosilla mentioned, there's only one language spoken in The Netherlands, Belgium and Suriname. The only odd thing is that there is a term for the group of dialects spoken in Flanders, but there is no term for the group of dialects spoken in the Netherlands.
It's fine to say Ik ben een Vlaming en ik spreek Vlaams just as it's fine to say Ik ben een Brabander en ik spreek Brabants.