The word "Great" has many meanings, both in English and Chinese. It seems that Google Translate picks "magnificent", "grand" as its translation in this case.
While "Hope you'll have a great day" is said time and time again in English conversation, it's less so in Chinese - but it does not mean people do not say it.
There are a few different ways to say it depending on the context.
When someone is having a day out, for example, visiting a place, going to a theme park, going shopping with friends and so on, you can say:
祝你(今天)玩得愉快/祝你(今天)玩得愉快 = Hope you'll have fun./Have fun!
"Today" is often omitted in speaking here as it's a "day out" that happens today.
When someone is organising an event, staging a gig, hosting a fair or similar stuff, you can say:
祝你今天順利/祝你今天顺利 = Hope it goes well today/Hope your gig goes well today.
"Today" is usually not omitted here in speaking as it emphasises the event is happening today.
When you say it to a business owner or someone running a shop, this can translate into:
祝你今天生意興隆/祝你今天生意兴隆 = Hope you'll have good business today.
Which according to Google Translate, it's "I wish you business is booming today."
I'm sure there are other occasions where you'd say "Hope you'll have a great day" and it is not the same Chinese expression. If anyone can think of any, please share with us :)