Finding optimal designs for generalized linear models is a challenging problem. Recent research has identified the structure of optimal designs for generalized linear models with a single or multiple independent explanatory variables that appear as first-order terms in the predictor. We consider generalized linear models with a single-variable quadratic polynomial predictor under a popular family of optimality criteria. When the design region is unrestricted, our results establish that optimal designs can be found within a subclass of designs based on a small support with symmetric structure. We show that the same conclusion holds with certain restrictions on the design region, but in other cases, a larger subclass may have to be considered. In addition, we derive explicit expressions for some $D$-optimal designs.