Both results indicate the both integrals result in zero.

The integrand is dominated by the part `Cosh[(ArcSinh[a] - ArcSinh[b])/2]*Exp[-1/2*(a^2 + b^2)] `

 Plot3D[ Cosh[(ArcSinh[a] - ArcSinh[b])/2]*Exp[-1/2*(a^2 + b^2)] 
 , {a, -10 , +10 }, {b, -10 , +10 }, 
 PlotPoints -> 100, PlotRange -> All, AxesLabel -> {a, b}] 

[![enter image description here][1]][1] 

For numerical reasons it is sufficient to decrease the integration range accordingly.

 inf=10; 
 f[x_, y_] :=NIntegrate[
 Cosh[(ArcSinh[a] - ArcSinh[b])/2]*Exp[-1/2*(a^2 + b^2)]*
 Cos[x*(a - b) +y*(Sqrt[1 + a^2] -Sqrt[1 + b^2])], {a, -inf, +inf}, {b, -inf, +inf}, Method -> "LocalAdaptive" , 
 IntegrationMonitor :> ((errors = Through[#1@"Error"]) &)];

 f[20, 10 ] (* result 1.86704*10^-45*)
 Total@errors (* error 1.93674*10^-46*)

Hope it helps!



 


 [1]: https://i.sstatic.net/Oumxp.png