Where was the first official weather forecasting office or departement established
under Governmental auspices?

Below from National Oceanic and Atmospheric Administration's National Weather Service



The beginning of the National Weather Service we know today started on February 9th, 1870, when President Ulysses S. Grant signed a joint resolution of Congress authorizing the Secretary of War to establish a national weather service. This resolution required the Secretary of War "to provide for taking meteorological observations at the military stations in the interior of the continent and at other points in the States and Territories...and for giving notice on the northern (Great) Lakes and on the seacoast by magnetic telegraph and marine signals, of the approach and force of storms"

After much thought and consideration, it was decided that this agency would be placed under the Secretary of War because military discipline would probably secure the greatest promptness, regularity, and accuracy in the required observations. Within the Department of War, it was assigned to the Signal Service Corps under Brigadier General Albert J. Myer. General Meyer gave the National Weather Service its first name: The Division of Telegrams and Reports for the Benefit of Commerce.

Ulysses S. Grant

This agency was placed under the Secretary of War because"military discipline would probably secure the greatest promptness, regularity, and accuracy in the required observations." Within the Department of War, it was assigned to the Signal Service Corps under Brigadier General Albert J. Myer. General Meyer gave the National Weather Service its first name: The Division of Telegrams and Reports for the Benefit of Commerce.

Later that year, the first systematized, synchronous weather observations ever taken in the U.S. were made by "observing-sergeants" of the Army Signal Service at 22 stations and telegraphed to Washington. An agency was born which would affect the daily lives of most of the citizens of the United States through its forecasts and warnings.

Of course, this just pertains to the United States.

According to Founders of Meteorology:

"...the British Government decided to set up a Meteorological Office and appointed (Robert) FitzRoy to be its director."

This site gives us:

"Admiral Robert Fitzroy established the first weather warning service in 1859. Before his promotion to admiral, he spent many years as a sea captain, including five years at the helm of the Beagle, which took Charles Darwin on his famous voyage. From his years at sea, Fitzroy knew the importance of weather conditions. With the arrival of the telegraph, he was able to obtain reports from remote stations. The telegraph reports were received and posted at Lloyds of London, a coffee house where ship captains and sailors gathered before sailing."

I'm guessing that the 1859 date corresponds with the date the British government set up Fitzroy's office. Whatever the case, Fitzroy died several years before the establishment of the United States' National Weather Service.

Anything prior to this? Surely the ancient Greeks or Chinese must have given it a shot!

I found references to both the Greeks and the Chinese, Fuse, but not tied in with any type of governmental office. Surely the Chinese, who correctly predicted eclipses before Columbus sailed, had a weatherman.

Check this out from NOAA Legacy Time Line

1807 - Survey of the Coast established by President Thomas Jefferson. This was to become America’s first physical science agency. Ferdinand Hassler’s plan, based on scientific principles, was selected by Government as the best of many presented.

1814 - Surgeon General orders surgeons to keep weather diaries; first government collection of weather data

1842 - James P. Espy appointed first official U.S. Government meteorologist

The bill that Ulysses Grant signed in 1870 established a national weather warning service.

"Consideration and analysis of The Five Elements, " Dao " - Moral Unity, " Tian " - Weather Condition, " Di " - Geographical Condition, " Jiang " - Leadership Quality, " Fa " - Discipline and Organisation Structure, a must know for all commanders. Victory to those who understand and no victory to those who does not. The Five Elements will determine success or failure of conducting war."
- Sun Tzu (circa 500 BCE)

According to Marco Polo, Kublai Khan ((1215-1294) maintained some 5,000 court astrologers, whose duties included the hazardous task of weather prediction.

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"Astronomy was a vital science in ancient China, when it was used to predict the weather." - China Outdoors
Kublai also set up an observatory.