This is a direct extract from the book 'Cronicle of Aviation'
©1992 Chronicle Publications, Editor in Cheif Bill Gunston
The Hague, October 7 1919
A new Dutch airline was formally registered here today. Its name is Koninklijke Maatschappij voor Nederland en Kolonien [Royal Air Travel Company for the Netherlands and Colonies], or KLM for short. Backed by powerful business and banking interests, KLM has a sound financial basis although it does not yet have any registered aircraft. It will be based at Amsterdam's Schipol airport.
KLM's able managing director, Albert Plesman, began planning for the airline a few months ago. It was originaly to be known as Nederlandsche en Koloniale Luchtverkeer Maatschappij [Dutch and Colonial Airline Company], but on September 12, Queen Wilhelmina authorized the use of the word royal.
May 17 1920
A new service between Amsterdam and London, to be operated jointly by KLM and the British company Aircraft Transport & Travel (AT&T), opens.
So, assuming this book is correct, and I'd assume it is, KLM was registered in 1919, and made its first flight in 1920.
"So powerful is the light of unity that it can illuminate the whole earth." - Bahá'u'lláh