P={{p11,p12},{p12,p22}}; (* must be symmetric *)
Phi={{phi11,phi12},{phi21,phi22}}; (* does not need to be symmetric *)
L={{lam11,lam12},{lam21,lam22}}; (* does not need to be symmetric *)
Q={{q11,0},{0,q22}}; (* must be symmetric and we take it to be diagonal *)

OneUpdate =Phi . P . Transpose[Phi]+L . Q . Transpose[L]//Simplify

(* extract the components (1,1), (1,2), and (2,2) *)
OneUpdate[[1,1]]
OneUpdate[[1,2]]
OneUpdate[[2,2]]
OneUpdate[[1,2]]==OneUpdate[[2,1]]