A. Gilda comes out to see Arimaspi dead alongside the guards and a dying Guto. She rushes to him and summons aid, but it is too late. Before he dies, Guto appoints Gilda as his successor, making her promise to keep the idol and kingdom safe, which she does. Once he dies, Gilda ascends to the throne and becomes Queen of Griffonstone, intent on keeping Guto's promise
B. Gilda comes out to see Arimaspi dead, with the guards securing the area and Guto alive but a little banged up. Guto thanks Gilda for the warning and saving the idol. Now two interesting things could happen here: 1A. Guto offers Gilda riches beyond her wildest dreams 2B Guto offers his claw to Gilda, asking her to marry him
For me personally, I like to go with the second scenario and 2B. Now of course, Guto would be in disbelief to know that Gilda is from the future, but would be interested in preventing the decline of the kingdom, which Gilda would be happy to share the information, in exchange of being granted her own bakery. I imagine that Gilda would be loved by the other griffons.