I have done a few (not extensive) numerical tests that led me to this conjecture:
$$2m+1 \text{ is prime} \iff \exists n:\frac{F_{n}^{2m+1}+F_{n+1}^{2m+1}}{F_{n}+F_{n+1}} \text{ is prime}$$
where $m \ge 1$ and $n \gt 1$.
Therefore, according to the conjecture, if $2m+1$ is composite then $\frac{F_{n}^{2m+1}+F_{n+1}^{2m+1}}{F_{n}+F_{n+1}}$ is composite too for any $n \gt 1$. However, I can't see an easy application of known identities to prove this or the full conjecture.
Any idea?
UPDATE 1 (generalization)
This might not be related to Fibonacci numbers, because it seems to work also with e.g. $\frac{(2n+1)^{2m+1}+(3n+1)^{2m+1}}{5n+2}$ so there might be a generalization. The generalization could be: ${2m+1} \space \text{is prime} \space \iff \exists a,b \space\text{coprime} \space : \frac{a^{2m+1}+b^{2m+1}}{a+b} \space\text {is prime}$.
UPDATE 2 (bound for $n$)
Maybe the conjecture could be simplified with an upper bound for $n$, although I don't have any idea which bound could be good.
UPDATE 3 (easier direction proved)
If $2m+1=uv$, $u,v \gt 1$, then $\frac{a^{uv}+b^{uv}}{a+b}=\frac{a^{uv}+b^{uv}}{a^u+b^u}\frac{a^{u}+b^{u}}{a+b}$, where both fractions on the RHS are integers greater than $1$, therefore $\frac{a^{uv}+b^{uv}}{a+b}$ cannot be prime.
UPDATE 4
The more difficult direction will probably remain a conjecture, since there already exist conjectures of this kind e.g.:
$n \space \text{is prime} \implies \exists b: \frac{b^n+1}{b+1} \space \text{is prime}$
(see OEIS A103795).
