You have the Gamma (with a capital G) function twice in your code, but in the image it is a lower case gamma. What is lower case gamma?
If you mean for gamma to be the incomplete Gamma function, then the closed form doesn't exist. However, with a suitable scaling, it is fairly easy to find numerical values for the integral:
Plot[With[{m = q},
NIntegrate[Exp[-y] Gamma[1 + m, 0, Sqrt[y]], {y, 0, Infinity}]], {q,
0, 10}]
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