Meaning, why can't this exist:
Is this a calculus question? Or is this just by the definition of current?
It can exist, and I would consider this a calculus question.
When Newton and Leibniz originally invented calculus, they conceived of a derivative as a (usually finite) ratio of two infinitesimal numbers. Infinitesimal means smaller in absolute value than any positive real number, but greater than zero. This was the way people did calculus for about 150 years. During the 19th century, mathematicians decided to redo the foundations of calculus using limits rather than infinitesimals. Ca. 1960 it was shown that there was actually nothing wrong with the infinitesimal approach. There is a nice online book by Keisler http://www.math.wisc.edu/~keisler/calc.html that does freshman calc using an updated version of the infinitesimal approach. There is nothing particularly hard or scary about any of this, and modern historical scholarship has shown that Leibniz basically had the whole system figured out correctly in many ways, even by modern standards of rigor: Blaszczyk, Katz, and Sherry, Ten Misconceptions from the History of Analysis and Their Debunking, http://arxiv.org/abs/1202.4153
In the Leibniz notation, d just means "a little bit of..." (where "little" means "infinitesimal"). Your equation $dI=dq/dt$ makes perfect sense. It simply says that a little bit of charge flowed over a little bit of time, and the result was a little bit of current.
There is a notational issue, which is that although we often think of the derivative $q'(t)$ as being the same as the ratio of the infinitesimals $dq/dt$, actually if you want to be consistent you have to define the derivative as the standard part of that ratio. The standard part of a number is the closest real number to that number. In your equation, if division of the numbers $dq$ and $dt$ gives an infinitesimally small result, then the standard part of $dq/dt$ is zero, so the derivative $q'(t)$ is zero. So there is a distinction between a number and its standard part. Leibniz was in fact careful about this distinction and had a special notation to represent it, but later writers felt that it was too cumbersome and stopped writing it. When infinitesimals were rehabilitated in the 1960s, they had to reinvent this distinction, and they invented the term "standard part."
There are certain grammatical rules for manipulating infinitesimals, but your equation doesn't violate any of them. For example, it would be grammatically incorrect to write
$$ du=1/dv .$$
This is because if you take a finite number divided by an infinitesimal number, you are guaranteed to get an infinite number, but a notation like $du$ is used to represent an infinitesimal number, not an infinite one.
Your equation $dI=dq/dt$ doesn't violate these rules because it's perfectly possible to divide an infinitesimal by another infinitesimal and get a result that is infinitesimal rather than finite. For example, suppose that $dx$ is a nonzero infinitesimal. Then we have as an identity $dx=dx^2/dx$.