One thing to try is do a change in variable to get rid of y.
Thank you Frank for your reply.
I will try with t=y-x to see how it goes. Does it sound good?
Best regards.
Might be better to set t = x - y to keep the terms in exponents negative.
Thank you for the valuable suggestion. I will do it.
I could not find symbolic integral with erf function.
a = 1;
b = 2;
y = 3;
NIntegrate[Sqrt[x] Exp[-x/a] Exp[-(y - x)^2/b], {x, a, b}]
(*0.0867826*)
where:
1/2 b E^((b - 4 a y)/(4 a^2)) Sqrt[\[Pi]] Erf[(b + 2 a b - 2 a y)/(2 a Sqrt[b])] -
1/2 Sqrt[a b] E^((b - 4 a y)/(4 a^2)) Sqrt[\[Pi]]Erf[(2 a^2 + b - 2 a y)/(2 a Sqrt[b])] -
1/2 Sqrt[b] E^((b - 4 a y)/(4 a^2)) Sqrt[\[Pi]]*NIntegrate[Erf[(b/a + 2 t^2 - 2 y)/(2 Sqrt[b])], {t, Sqrt[a], Sqrt[b]}]
(*0.0867826*)
Update:
Substituting x = t+y, t = a s-b/(2 a), s = m sqrt(b)/a and Integrating by parts:
but the last integral MMA and Maple can’t find.
If we simplify last integral to:
Integrate[Erf[m]/Sqrt[m + 1], m]
(* Integrate[Erf[m]/Sqrt[m + 1], m]*)
it is impossible to find antiderivative
The error function probably came from integration by parts.
Thank you Mariusz for the reply.
I have found that the integration is achievable if we get rid of sqrt(x). So I was thinking to use t=sqrt(x). Does it sound correct?
Yes. Error function is common for these kinds of integration.
I tried, but nothing have got. 
Thank you for your efforts..
Rubi is also unable to provide an antiderivative.
Int[Sqrt[x] Exp[-x/a] Exp[-(y - x)^2/b], x]
is returned unevaluated.