Technically yes you can start a sentence with a conjunction like "but" or "and" although it should be done infrequently. However, in this sentence, it seems wise just to delete the period and add a comma. During his younger years, Travis Rowley had been through many ups and downs with his family, school, and even friends. But, as soon as Deena arrives, things started to change As my children have found when they ask me if they can do something, the answer is yes. They have then gotten in trouble, to which they respond, you said I could. My response to their statement is yes, I told you that you could, but I never gave my permission, as you never used the phase "May I".
Yes, you can start a sentence with but, but should you, in general no.