In this work, we have obtained maximal values for atmospheric mixing angle and $CP$ violating Dirac phase of the lepton sector in a type II seesaw scenario, by modifying a model originally proposed by Grimus and Lavoura, where we have used $CP$ and other discrete symmetries. In order to make predictions about neutrino mass ordering and the smallness of the reactor angle, we have obtained some conditions on the elements of the neutrino mass matrix of our model. Within the same framework, we have studied quark masses and mixing by using a certain texture in a higher-dimensional model.