Is there anything that is known about what is the maximal number of incidences between quadratic forms and points? I looked at the internet and I haven't found anything that works for something that is not a sphere or has a dimension more than 3, but since I have rather "simple" kind of quadratic forms, I wondered if anything is known about such forms. Basically, if we take $6$ variables $x_1,.....,x_6$, then the quadratic forms are of the form $a(x_2x_6x_3x_5)+b(x_1x_6x_3x_4)+c(x_1x_5x_2x_4)=k$. I wondered if there is a literature where such form come as a special case of some theorem.

Your terse question leaves many possible interpretations, and forgive me if this is a misinterpretation. The 2012 paper by Micha Sharir, Adam Sheffer, and Joshua Zahl,
showed that the number of incidences between $m$ points and $n$ arbitrary circles in three dimensions—in the situation when the circles are "truly threedimensional," in the sense that there exists a $q < n$ so that no sphere or plane contains more than $q$ of the circles—then the number of incidences is $$O^*\big(m^{3/7}n^{6/7} + m^{2/3}n^{1/2}q^{1/6} + m^{6/11}n^{15/22}q^{3/22} + m + n\big).$$ The complexity of this result illustrates the complexity of the topic!



Well, it seems the Kővári–Sós–Turán theorem should give you a bound on the number of incidences, though this bound is likely not sharp. The Kővári–Sós–Turán theorem says that if you have a collection of surfaces with the properties that: 1) Any $t$ surfaces have at most $O(1)$ points common to all $t$ of them Then if you have $m$ points and $n$ such surfaces, the total number of incidences is $O(\operatorname{min}(mn^{1/s},\ m^{1/t}n))$. 

