The modeling of gene networks from transcriptional expression data is an important tool in biomedical research to reveal signaling pathways and to identify treatment targets. Current gene network modeling is primarily based on the use of Gaussian graphical models applied to continuous data, which give a closed-form marginal likelihood. In this paper, we extend network modeling to discrete data, specifically data from serial analysis of gene expression, and RNA-sequencing experiments, both of which generate counts of mRNA transcripts in cell samples. We propose a generalized linear model to fit the discrete gene expression data and assume that the log ratios of the mean expression levels follow a Gaussian distribution. We restrict the gene network structures to decomposable graphs and derive the graphs by selecting the covariance matrix of the Gaussian distribution with the hyper-inverse Wishart priors. Furthermore, we incorporate prior network models based on gene ontology information, which avails existing biological information on the genes of interest. We conduct simulation studies to examine the performance of our discrete graphical model and apply the method to two real datasets for gene network inference.