You can use Solve with a 3rd argument specifying which variables to eliminate:

Solve[
    {
    Subscript[NE,t]==Subscript[X,t]/(γ*σ^2),
    Subscript[NF,t]==(Subscript[D,t]-(T-t-1)*γ*σ^2*Q-Subscript[P,t])/(γ*σ^2),
    uf==1-ue,
    Q==ue*Subscript[NE,t]+uf*Subscript[NF,t]
    },
    Subscript[P, t],
    {uf, Q, Subscript[NE, t]}
]

{{Subscript[P, t] -> -γ σ^2 (-(Subscript[D, t]/(γ σ^2)) + Subscript[NF, t] - (-1 - t + T) ((-1 + ue) Subscript[NF, t] - ( ue Subscript[X, t])/(γ σ^2)))}}

You can use Solve with a 3rd argument specifying which variables to eliminate:

Solve[
    {
    Subscript[NE,t]==Subscript[X,t]/(γ*σ^2),
    Subscript[NF,t]==(Subscript[D,t]-(T-t-1)*γ*σ^2*Q-Subscript[P,t])/(γ*σ^2),
    uf==1-ue,
    Q==ue*Subscript[NE,t]+uf*Subscript[NF,t]
    },
    Subscript[P, t],
    {uf, Q, Subscript[NE, t]}
]

{{Subscript[P, t] -> -γ σ^2 (-(Subscript[D, t]/(γ σ^2)) + Subscript[NF, t] - (-1 - t + T) ((-1 + ue) Subscript[NF, t] - ( ue Subscript[X, t])/(γ σ^2)))}}

Addendum

If you eliminate ue, Subscript[NE,t] and Subscript[NF,t], then after a simple replacement you will arrive at the solution from your paper:

Solve[
    {
    Subscript[NE,t]==Subscript[X,t]/(γ*σ^2),
    Subscript[NF,t]==(Subscript[D,t]-(T-t-1)*γ*σ^2*Q-Subscript[P,t])/(γ*σ^2),
    uf==1-ue,
    Q==ue*Subscript[NE,t]+uf*Subscript[NF,t]
    },
    Subscript[P, t],
    {ue, Subscript[NF, t], Subscript[NE, t]}
] /. Subscript[X, t] - uf Subscript[X, t] -> ue Subscript[X, t]

{{Subscript[P, t] -> (-Q γ σ^2 + Q uf γ σ^2 + Q t uf γ σ^2 - Q T uf γ σ^2 + uf Subscript[D, t] + ue Subscript[X, t])/uf}}