There is no concrete evidence to suggest that GTA 5's Franklin is the son of CJ from GTA San Andreas.
Regardless, many GTA 5 fans look up this question. CJ isn't present in GTA 5 (in an official capacity), and he doesn't have many similarities to Franklin. There is nothing to suggest that CJ is related to Franklin in any way.
The GTA series is no stranger to absurd theories, but it's worth looking at both sides of the argument for the sake of a discussion.
CJ is the main protagonist of GTA San Andreas, whereas Franklin is the main protagonist of GTA 5. Gamers never find out who Franklin's father is, although they do know that he's dead.
Is GTA 5's Franklin the son of GTA San Andreas's CJ?
Considering there is no definitive proof one way or the other, it's vital to look at all of the evidence that players have before them. CJ's voice actor (Young Maylay) is the younger cousin of Franklin's voice actor (Shawn Fonteno).
This real-life connection is the only valid connection between the two characters. Otherwise, everything else is based on speculation and hearsay.
Franklin's birth year
Franklin Clinton was born in 1988. GTA San Andreas took place in 1992, and CJ isn't known to be a father by this time. Of course, GTA San Andreas is part of the 3D universe, so it doesn't have to follow the lore of the HD universe.
However, one could argue that because they're a part of two different canon universes, it doesn't mean there would be any connection between the two characters.
3D vs. HD universe
Rockstar Games has three "universes" within the GTA series. The 2D, 3D, and HD universe are the three, but the 2D universe isn't relevant to this topic. CJ comes from the 3D universe, and GTA 5's Franklin comes from the HD universe.
These "universes" count as separate lores. What happens in GTA San Andreas doesn't necessarily occur in GTA 5's timeline. There are only a few characters present in both universes, and CJ isn't one of them.
Hence, it's far more likely that Franklin's father is just some random, unnamed person than it is for CJ to be Franklin's dad. Some players even speculate that GTA San Andreas's Denise Robinson is GTA 5's Denise Clinton, but there is nothing to suggest that's the case.
Other notes
There are also theories that CJ could be Franklin's uncle. However, these rumors are as unsubstantiated as the "CJ is Franklin's dad" rumors. Both protagonists live in San Andreas, but that's a flimsy reason for any correlation between the two characters.
There's hardly any connection between GTA 5 and GTA San Andreas. A large part of this lies in the fact that they belong to two separate "universes." The HD universe comprises of GTA 4 and its DLC episodes, GTA Chinatown Wars, GTA 5, and GTA Online.
GTA San Andreas is not a part of the HD universe. Likewise, there aren't any genuine hints in GTA 5 that suggest that Franklin is related to CJ. Anything associated with this theory is sheerly based on player speculation.