The question is confusing, and so maybe misinterpreted by some commenters and answerers. In your citation, there is two random variables $X$ and $Y$ which are independent. Then, there is a theorem saying that they are uncorrelated. It also have an easy proof, which you can find in many probability texts.
But this do not mean that if you have a sample $(X_1,Y_1), \dotsc, (X_n,Y_n)$ from $(X,Y)$, that the sample correlation coefficient will be zero! which is what the answer by @Nutle explains. But, if $n$ is large, the sampling distribution of that correlation coefficient will be concentrated close to zero.
So, yes, samples from two independent variables can seem to be correlated, by chance. Especially if $n$ is small. That just means that you risk having a type I error.