Wolfram Alpha is going to use Mathematica for the computation. The documented behavior for Mathematica is quite easy to find by web search. Here is the relevant link.
https://reference.wolfram.com/language/ref/FourierTransform.html
Click on Details and Options opener to see information on definition used and parameter settings.
Dear Daniel,
Many thanks for the quick reply! Sorry for the stupid question. I see now, this is not a bug, the exponent in the Fourier transform is simply exp(+iwt), not exp(-iwt) as I assumed!
Best regards, /Alexey