I have a set of numbers generated from Gamma distribution. How can I verify that these values were all randomly generated from Gamma or satisfy Gamma distribution?
You can't say "these are definitely from a gamma distribution" by looking at the data.
They might, for example, be from something that's close to, but not, a gamma distribution. (In fact, real data could almost never be exactly gamma, so it's pretty much a waste of time to check that even if you could. All models are wrong, but some are useful, as Box put it.)
You can hope to identify when the data are not gamma distributed. That's what goodness of fit tests do, for example.
However, if your question is really more along the lines of "is a gamma model for my data a close enough approximation that the model is useful" (for whatever 'useful' might mean for your purposes), then such a hypothesis test doesn't really relate to that*, and it might be better answered by assessing the extent to which (and form of) deviation from gamma that you have.
* however you might be perhaps able to pursue the notion of something like an equivalence test for this sort of purpose.
You might use simulation, or even resampling to partially assess the extent to which the sort of deviation from gamma you might have would affect your results. (Such answers will tend to be somewhat vague compared to the rather black and white outcome of a hypothesis test, but Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise, as Tukey said.)