The substitution suggested by Maxim also appears to work for the zero-th order functions
gr:=(beta,r)->(1/8*I)*(-HankelH1(0, beta*r)-(2*I)*BesselK(0, beta*r)/Pi)/beta^2;
p:=subs(s = sqrt(tau), (1/2)*tau^(-1/2)*convert(cos(alpha*s)*gr(beta, s), MeijerG, include = cos));
Check analytical result
Interestingly, Maple evaluates the integral in general before the one applies the assumptions. Trying to help it by introducing earlier the assumptions about alpha and beta ie. move the assumption line to just above the definition of q and Maple fails to find the integral for the special case alpha>0 and beta>0.