The functional quadratic regression model assumes a polynomial, rather than linear relationship between the scalar response variable and a functional predictor variable. This paper focuses on the statistical inference tailored for the functional quadratic expectile regression model. Functional coefficients are approximated by the functional principal component basis functions, and asymptotic properties of estimators are derived under some mild conditions. Furthermore, to inspect the effect of the functional quadratic term on the response variable, we develop an expectile rank score test and establish its asymptotic property. Simulations are conducted to assess the empirical performance of the proposed estimation methods and test statistic. Results indicate that the proposed estimators are comparable to competing estimation methods and the newly proposed expectile rank score test is effective. Finally, the advantages of our methodologies are illustrated using a real-data example.