In this paper, we consider fluid queue models with infinite buffer capacity in which the fluid flow is governed by a birth and death process (BDP) with quadratic arrival and service rates on a finite state space. We use certain interesting identities of the tridiagonal determinants to analytically determine the eigenvalues of the underlying tridiagonal matrix that governs the probabilistic behaviour of the system and hence obtain the stationary buffer content distribution of the process. Numerical investigations are presented in the form of graphs to capture the variations in the behaviour of buffer content distribution.