There are several candidates for this honour, depending on what we mean by “watch”.
As well as anonymous cave-painters of roughly 18,000 years ago, and Noah, who sent forth a raven and dove from the Ark, there is the scientist and philosopher Aristotle (384-322 BC), who made many accurate (and a few inaccurate) observations of birds. If we regard birdwatching as being the observation of birds primarily for pleasure, as distinct from study, then probably the Reverend Gilbert White, Vicar od Selborne in Hampshire (UK). In 1789 he published one of the best-known books in English “The Natural History of Selborne” . This book was a compilation of 44 of his letters to Thomas Pennant, the leading British zoologist of the day, and 66 letters to the Hon. Daines Barrington, an English barrister and another Fellow of the Royal Society. In these letters White detailed the natural history of the area around his family home at the vicarage of Selborne in Hampshire.