Why is the Hilbert curve called like that? Everywhere the function $H\colon\mathcal{I}\to Q=[0,1]\times[0,1]$ is called the Hilbert curve. Is this a historical thing because normally one would call $H\colon\mathcal{I}\to Q$ maybe the Hilbert path and $H(\mathcal{I})$ the Hilbert curve.
Right now I am writing my bachelor thesis and I am unsure wether I should explain the term "Hilbert curve" as the function itself and not the image which would be just $[0,1]\times[0,1]$.

In some contexts people are careful to distinguish between a function and its graph, while in others they aren't.  Moreover, even in the former circumstance, which of the two is known as the "curve" also varies according to context.  In topology a "curve" is typically a function $\gamma : I \to X$, whereas in geometry that function would be called a "parameterization" and its image would typically be called a curve.
The upshot is that you can typically count on definitions to be consistent within a particular paper or book, but not across mathematics in general.  This may not be an ideal state of affairs, but it's what we have to work with.
What this means for your talk is that you can call the two whatever you prefer but you should take care to explain the convention you're using if it's important to the talk.
