No, the two concepts are quite different.
If you have a process with a mean you are not so much interested in, you can predict the variance. An example is (G)ARCH modeling in financial applications, where we typically assume (by the efficient markets hypothesis) that the mean follows a random walk, and our interest shifts to predicting the volatility around this mean, i.e., the variance.
On the other hand, when you are predicting (and regardless of whether you are predicting the mean, or the variance, or something else!), if you re-run your prediction algorithm on a new sample of the underlying data generation process, you will get a different point prediction. That is, your prediction will vary. It will have a variance. And you can try to analyze the variance of your prediction. We typically want lower variance of predictions, because it is one component of the prediction error.
So, to tie the two concepts together, you might be interested in a prediction of the variance of a process... and then also in the variance of this prediction, i.e., in the variance of your variance prediction. (To take it one step further, this is again a prediction. A prediction of the variance of your (original) variance prediction.)