Sorry, I am no expert in special functions and integration.

Written this way we get Hypergeometric functions:

$Assumptions = \[CapitalTheta] > 0 && t > 0 && \[Delta] > 0 && 
   s < \[CapitalTheta];
Subscript[m, d][t_] = 
  Subscript[\[Lambda], 
     2] t + (Subscript[\[Lambda], 1] - Subscript[\[Lambda], 2])/
     Subscript[\[Lambda], 1] (1 - E^(-Subscript[\[Lambda], 1] t));
Subscript[M, X][s_] = 
 Integrate[
  E^(s x) \[CapitalTheta]*E^(-\[CapitalTheta] x), {x, 0, Infinity}]
Subscript[M, Subscript[Z, 0][t_]][s_] = 
 E^(Subscript[\[Lambda], 2] Integrate[
     Subscript[M, X][s E^(-\[Delta] v)], {v, 0, t}])

Integrate[(Subscript[M, X][s E^(-\[Delta] v)] - 1) Subscript[M, 
    Subscript[Z, 0] [t - v]][s E^(-\[Delta] v)] D[Subscript[m, d][v], 
   v], v]

I suspect it may be confused by the

D[Subscript[m, d][v],v]

at the end of your integral.

While you might mathematically understand what the integral of ln(x) dsin(x^3) means, I think if you look at all the documentation and examples I think you will see they all show integration with respect to a simple variable, not a function and not a subscripted variable.

Can you express your integral in such a form and try evaluating the integration again?