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With the help of the very handy TransLink journey planner, it turns out it can be done just fine!
Currently, there are no trains serving the Sunshine Coast (though a branch line is proposed). Instead, you need to take a Nambour Train from Roma Street Station or Central Station, and then change at somewhere like Woombye for the bus to the coast.
The train takes about 2 hours from Brisbane, then it's about another 45 minutes on the bus to Mooloolaba. There's roughly one train an hour, and the last southbound train leaves Nambour just after 8pm on a weekday. So, it is possible to leave Brisbane in the morning, spend the afternoon on the Sunshine Coast, then head back in the evening, all by public transport.
Oh, and just to note, sometimes off-peak the "train" between Nambour and Caboolture is actually a bus due to a lack of track capacity, which is a little bit slower.