heff[β_,p_,q_,m_,j_,s_,z_] := -β j(p-q) s^2-(m + j Sqrt[q]z)s;
phi[β_,p_,q_,m_,j_,z_,k_]:=((heff[β,p,q,m,j,1,z]+(-1)^k heff[β,p,q,m,j,-1,z])/(heff[β,p,q,m,j,1,z]+heff[β,p,q,m,j,-1,z]));
funcM[β_,p_,q_,m_,j_]:=NIntegrate[(1/(Sqrt[2 pi]))Exp[((-z^2)/2)]phi[β,p,q,m,j,z,1],{z,-Infinity,Inixed_pointfinity}];
funcP[β_,p_,q_,m_,j_]:=NIntegrate[(1/(Sqrt[2 pi]))Exp[((-z^2)/2)]phi[β,p,q,m,j,z,2],{z,-Infinity,Infinity}];
funcQ[β_,p_,q_,m_,j_]:=NIntegrate[(1/(Sqrt[2 pi]))Exp[((-z^2)/2)](phi[β,p,q,m,j,z,1])^2,{z,-Infinity,Infinity}];
β = 1.1;
j = 0.9;
FixedPoint[
{funcM[β,#[[1]],#[[2]],#[[3]],j],funcP[β,#[[1]],#[[2]],#[[3]],j],funcQ[β,#[[1]],#[[2]],#[[3]],j]} &,
{0.5,0.3,1.0}
]