The year when King Charles XII of Sweden was advancing towards Moscow during the great Northern War was exceptionally cold. This winter was not calculated by Sweden and Charles had to escape to the Ukraine and Ottoman Empire to resupply.
What were the causes of the Great Frost of 1709 and was it ever predicted?
Per the wikipedia page I edited into your answer, the causes were unknown, and it is still inexplicable using modern climate models, so even if someone had predicted it it would not have been based on anything other than guesswork.
This winter event has drawn the attention of modern-day climatologists in the European Union's Millennium Project because they are presently unable to correlate the known causes of cold weather in Europe today with weather patterns documented in 1709. According to Dennis Wheeler, a climatologist at the University of Sunderland: "Something unusual seems to have been happening".
Modern climate models do not appear to be entirely effective for explaining the climate of 1709.
Supporting sources from the wiki pages:
Adding to this, intuitively it makes sense to assume that the cause would have been some kind of polar vortex coming down onto Europe or its whereabouts. Usually this introduces pockets of cold coming from Siberia into Europe brought by winds from the (North-)East, as happened during the nastier winters we had in the past few years. The very odd part about the 1709 winter is that winds were reportedly coming from the West and the South.
I support @DenisdeBernardy's answer but I hope I can provide some additional helpful notes.
Firstly, a forecast in the modern sense could not have been made before starting to think about meteorology as a science and weather as something to be forecast based on observations. I take the view that your question does not ask whether a doom-monger said that "this coming winter will be our end"/"winter is coming" but rather whether a reasoned argument was made beforehand on that the winter of 1709 would be difficult.
This could not have been so because though stand-alone observation posts had been set up by the mid-17th century, weather analysis techniques really only developed in the late 19th century:
The use of weather charts in a modern sense began in the middle portion of the 19th century. Weather map pioneers include William Charles Redfield, William Reid, Elias Loomis, and Sir Francis Galton, who created the first weather maps in order to devise a theory on storm systems. The invention of the telegraph in 1837 made it possible to gather weather information from multiple distant locations quickly enough to preserve its value for real-time applications.
It was not until the invention of the electric telegraph in 1835 that the modern age of weather forecasting began.
Weather lore, on the other hand, ("warm summer, warm winter", etc) is either too specific to be of value on a continent-scale or not proven to be correct.
Secondly, I would counter your suggestion that Karl XII moved into the Ukraine because of the winter. The move south was related to the tactical situation at the time, and while Englund's 'Poltava' mentions of a hard winter, this is not an incentive for the moves but rather a common obstacle for all warring parties:
The summer was warm and rainy. The soldiers had to exert themselves. Rye [probably mistranslated the grain] grew slowly and that meant raw grain had to be harvested… The Russians scorched the area between the Swedish army and Smolensk in great detail… The race into the Ukraine began. The Swedes and the Russians went south fast as possible: both wanted to gain control over the fertile territory in as great an extent as possible.
-Englund, 'Poltava', free translation from Estonian
… on the day following the news of Lesnaya the King wrote to Mazeppa promising his support and requesting the provision of winter quarters in the Ukraine. With a friendly populace and plentiful supplies, the Ukraine was seen as 'a country flowing with milk and honey'. The plan was now to winter the army in the Ukraine and to attack Moscow from the south in the spring.
The winter of 1708/9 came early. It was to be the worst in living memory. Birds froze in the trees, and riders and horses died as one frozen mass. The only protection from the icy wind in the empty Ukrainian landscape were the few towns and villages, which became fought over with a ruthlessness brought on by the need to survive. Despite the conditions, both monarchs saw advantages in continuing the fighting.
Charles's main aim was to ensure the safety of his winter quarters by ejecting the Russians from the settlements west of the lliver Vorskla, which would give him a secure territorial boundary.
-Konstam, 'Poltava 1709'