Hello Imre,
I would do something like this (where you might want to be careful with the additional trick "?_view=frameless" in order to get more interactivity, since it also needs more cloud credits).
Rolf
Clear[f1, f2, q1, q2, \[Beta], \[Nu], m]; \[Nu] = 2.42; m = 5;
f1[z_, \[Beta]_] := (1 - (1/5)*(\[Nu]*(1 - \[Beta]))*(1 - z))^5;
f2[z_, \[Beta]_] := 1 - \[Nu]*\[Beta]*(1 - z);
q1[z_, \[Beta]_, \[Rho]_] := f1[z, \[Beta]]/(\[Nu]*(1 - \[Rho])) + 1 - 1/(\[Nu]*(1 - \[Rho]));
q2[z_, \[Beta]_, \[Rho]_] := f2[z, \[Beta]]/(\[Nu]*(1 - \[Rho])) + 1 - 1/(\[Nu]*(1 - \[Rho]));
dyn = Manipulate[Plot[{y, q1[y, \[Beta], \[Rho]]*q2[y, \[Beta], \[Rho]]}, {y, 0, 1},
BaseStyle -> {FontFamily -> "Helvetica", FontSize -> 16,
FontWeight -> "Plain"}], {\[Beta], 0., 100*0.0065},
{\[Rho], 0, 1 - 1/\[Nu]}, SaveDefinitions -> True]
CloudDeploy[dyn, CloudObject["dynamic.nb?_view=frameless"],
Permissions -> {All -> {"Read", "Interact", "Evaluate"}}]