F={{0,0},{1,-b}};
GQGT={{q,0},{0,0}};
P={{p11,p12},{p12,p22}};
H={{0,a}};

pDot=F . P+P.Transpose[F]+GQGT-(1/r)P.Transpose[H].H.P;
MatrixForm[pDot]

p12Sol=Solve[pDot[[1,1]]==0,p12]

pDot22=pDot[[2,2]]/.p12Sol//Simplify

(* we have two possible solutions for p22 for each equation *)
firstTwo=Solve[pDot22[[1]]==0,p22] 
secondTwo=Solve[pDot22[[2]]==0,p22]

allSol=Join[firstTwo,secondTwo]

(* Now p22 must be postive and real so only a few of these are valid solutions *)