We consider the problems of computing the average degree and the size of a given network in a distributed fashion and under quantized communication. More specifically, we present two distributed algorithms, which rely on quantized operation (i.e., nodes process and transmit quantized messages) and are able to obtain the exact solutions in a finite number of steps. During the operation of our algorithms, each node can determine in a distributed manner whether convergence has been achieved and correspondingly terminate its operation. For terminating the operation of our algorithms, we assume a known bound for the network diameter. To the best of the authors' knowledge, these algorithms are the first to find exact solutions (i.e., with no error in the final result) under quantized communication. Note that our network size calculation algorithm is the first in the literature to calculate the exact size of a network in a finite number of steps without introducing a final error; in other algorithms, this error can be either due to quantization or asymptotic convergence. In our case, no error is introduced since the desired result is calculated in the form of a fraction involving an integer numerator and an integer denominator. We demonstrate the operation of our algorithms and their potential advantages through simulations.