The quadrants are usually defined by means of the sign of its constituents. For example, the first quadrant is the set of ordered pairs $(x,y)$ such as both $x$ and $y$ are positive. So, your question is basically wheter a point such as $(c,0)$ can be considered both on the first and fourth quadrants or not. To my view, using the sign convention to define quadrants, this reduces to the question "is zero positive or negative?" I don't think there is a straight answer to this question and a discussion on this topic can be found here.

On the other hand, I don't think that this was the purpose of the original question. To my understanding, the question could be rephrased as "If $f(c) = 0$, can one conclude that $f$ is negative at some point, that is, does the graph of $f$ has a point (striclty) below the $x$-axis?" To answer this question, we can look for examples. Take $f(x) = |x-c|$, for instance. Then $f(c) = 0$ but what can you say about $f$ being negative? Again, this can only be answered by avoinding the sign convention of zero.