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We improve some previously known deterministic algorithms for finding integer solutions 
$x,y$ to the exponential equation of the form 
$af^{x}+bg^{y}=c$ over finite fields.
In this paper, we consider the Diophantine equations with positive integers 
$$\begin{eqnarray}\displaystyle F_{n}^{q}\pm F_{m}^{q}=y^{p} & & \displaystyle \nonumber\end{eqnarray}$$
$q,p\geq 2$ and 
$\gcd (F_{n},F_{m})=1$, where 
$F_{k}$ is a Fibonacci number. We obtain results for 
$q=2$ or 
$q$ an odd prime with 
$q\equiv 3\;(\text{mod}\;4),3<q<1087$, and complete solutions for 
$q=3$.
