To my knowledge the first Inland Rivers steamboat fitted with a calliope was the Packet/Str. Unicorn, at Cincinnati, OH. This was in 1860. The instrument was a Joshua Stoddard "American Steam Piano". It had both manual (keyboard) action and a spoked cylinder that when cranked, played a simple tune.
This comes from Steven Espensheid's book, the "The History of the Great American Steam Piano".
I doubt this was the "first" application on a riverboat. However, it is the first confirmation I have seen in print.
Many stories are floating around about calliope use on steam trains. Although, I have seen one newspaper account of this, that single press mention is the only I have seen.
The calliope did not make appearance in the circus until the 1870's, well after the Civil War.
There are stories also told of calliope whistles being used as mortar shell casings in the Civil War. I do not doubt this, as any steam whistle bell is perfectly suited for such use. As a mater of fact, the whistles Earnest J. Wilde made for the calliopes he built for the Johnson Party Barge (aka: BB Funliner and City of Memphis)and the Str. Belle of Louisville were built of spent mortar casings from WWII. So that story has come full circle!
However, the spent mortar shells did not hold up very well and only one of the Earnest J. Wilde instruments survives to this day. The Late Mr. Wilde's personal instrument, today is the property of Capt. Hal Wilmunder, owner of the Str. Elizabeth Louise in Sacramento, CA. It is on a Circus Wagon of sorts and is used at steam shows in the area.
I have once again rambled way beyond the point that I tried to make.
The first calliope placed on a river steamboat I can confirm is the Str. Unicorn, in 1860.