Disrupted or abnormal biological processes responsible for cancers often quantitatively manifest as disrupted additive and multiplicative interactions of gene/protein expressions correlating with cancer progression. However, the examination of all possible combinatorial interactions between gene features in most case-control studies with limited training data is computationally infeasible. In this paper, we propose a practically feasible data integration approach, QUIRE (QUadratic Interactions among infoRmative fEatures), to identify discriminative complex interactions among informative gene features for cancer diagnosis and biomarker discovery directly based on patient blood samples. QUIRE works in two stages, where it first identifies functionally relevant gene groups for the disease with the help of gene functional annotations and available physical protein interactions, then it explores the combinatorial relationships among the genes from the selected informative groups. Based on our private experimentally generated data from patient blood samples using a novel SOMAmer (Slow Off-rate Modified Aptamer) technology, we apply QUIRE to cancer diagnosis and biomarker discovery for Renal Cell Carcinoma (RCC) and Ovarian Cancer (OVC). To further demonstrate the general applicability of our approach, we also apply QUIRE to a publicly available Colorectal Cancer (CRC) dataset that can be used to prioritize our SOMAmer design. Our experimental results show that QUIRE identifies gene-gene interactions that can better identify the different cancer stages of samples, as compared to other state-of-the-art feature selection methods. A literature survey shows that many of the interactions identified by QUIRE play important roles in the development of cancer.