Well, the partition function is defined for positive integers, thus discrete and inherently one-dimensional. These properties makes it harder to create "nice fractal pictures" from this function.

However, the Collatz function, (multiply by 3 and add 1 if odd, divide by 2 if even), also have these properties. The Collatz function may however be analytically extended to an entire function, and iterating this function as when constructing a Julia set, DO create pictures one would expect.

99% of all fractals I've encountered are obtained by recursion somehow,
so you might need to find such recursion somehow in the computation for the theta function.
If this is possible, then you might be able to draw nice pictures.

I haven't watched the movie, but there seems to be some sequence which eventually becomes a multiple of 5. Now, this sounds like escape-time calculations to me, so if this can be extended in a continuous (preferably analytic) manner, then you might have a fractal.