Is it correct to say “Hope you’re having a great day”?

es, perfectly. It’s a shortened version of “I hope you’re having a great day”, with the “I” being understood. Other expressions using the same abbreviation are “Wishing you a happy birthday | anniversary | holiday” or “Sorry to hear about your loss | misfortune, etc. If you want to be scrupulously grammatical, you could write it as “ ’Hope you’re having a great day”, with the apostrophe before “hope” signifying a missing word or syllable—as in the original and correct spelling of the word “ ’bus” (the vehicle that carries many passengers) , shortened from “omnibus”.