That depends on what you specialize in. There’s little to no ancient history in and of itself involved, beyond discussing how certain events may have played a role in a language’s evolution. That falls under historical linguistics. Classics is really only involved insofar as languages like Greek and Latin are studied. You may choose to study an ancient language to supplement your linguistics program, but it’s certainly not required. I personally don’t know much about ancient languages, or historical linguistics for that matter. Unless you want to count one small Latin phonology problem I did in class, and one lecture on how modern English evolved 
Do you currently live in California? I go to UC Davis, myself. I can’t say I have personal experience with credentialing here to help much though, sorry. I do agree that it’s in your best interest to major in what you hope to teach, regardless.