I would first get an idea of where a solution might be using a plot:
eqs = {V2[
10, \[Sigma]] + (F11[
3/10, \[Alpha]]*(V1[40, \[Sigma]] - V2[10, \[Sigma]])) ==
V3[5, \[Sigma]] + (F22[
1/10, \[Alpha]]*(V3[150, \[Sigma]] - V4[5, \[Sigma]])),
V2[30, \[Sigma]] + (F11[
9/10, \[Alpha]]*(V1[40, \[Sigma]] - V2[30, \[Sigma]])) ==
V3[5, \[Sigma]] + (F22[
7/10, \[Alpha]]*(V3[68, \[Sigma]] - V4[5, \[Sigma]]))};
ContourPlot[Evaluate@eqs, {\[Alpha], -2, 2}, {\[Sigma], -2, 2}]
and then approximate it with an iterative method:
In[93]:= FindRoot[eqs, {{\[Alpha], 0.7}, {\[Sigma], 0.7}}]
Out[93]= {\[Alpha] -> 0.713928, \[Sigma] -> 0.636601}