If you build a Maker's quarter directly on a river tile, it will exploit only the Production bonus.
If you build a Farmer's quarter straight on a river tile it will exploit only the Food bonus.

You will either need to build a disctrict that exploits both Foor and Production (administrative centre, Baray) to build directly on a river or simply build along the river's banks. Take a look here :

Thanks all, I understand now.

Any bonus is always related to a tile that can produce it, either naturally (exploitation) or with a quarter producing this.

So, on river, it's better to build administrative center, a emblematic quarter giving both food & industry. Or if you are ready to only focus on one of the 2 bonus anything else. Maker or Food district, but you loose a bit. It can be ok if with the synergy bonus you get more at the end of the day.

If I understand well, the only bonus that apply to river in any case is the Tenet "+2 stability on river". Because any quarter get alway -10 stability, so it produce the ressource "stability", then the +2 bonus always happen. But imagine there is a quarter that don't provide stability (negative nor positive), so in this case, the river shouldn't produce stability.

It could be nice if this information could be on the wiki. 

Just a small tip that can be seen on the screenshot provided - you can still plant forests on river tiles and that can give you a 6 Food + 12 Production tile in the late game. You just need one Farmer's quarter + one Maker's quarter adjacent to the river tile to take advantage of both yields. Or simply a multiple yield district.

That makes for some juicy yields if you can make it work along the whole river.

Also not building districts on a river makes it eligible for + X on a tile producing Food/ on a river tile. The Harappans, Mexicans and Brazilians have such bonuses. Depending on your cultures and emblematic districts, those river tiles can be producing quite a lot of food an production.