For infinite fields of characteristic $p$, the Zariski open definition makes pretty much as much sense as it does in characteristic zero. Often, but not always, the same argument works.

Over finite fields, A good definition of "random" was provided by Bjorn Poonen in his paper Bertini Theorems over Finite Fields. Specialized to your case, it says that the proportion of tuples of polynomials $(f_1,\dots, f_6)$ over $\mathbb F_q$, of degrees $(d_1,\dots, d_6)$, which satisfy your condition, has positive lim inf as $d_1,\dots, d_6$ tend to $\infty$.

Note that it is too much to ask for in this setting that the probability tends to $1$ - for smoothness results, it almost never does, since one can force a singularity with a congruence condition mod $x_0^2$, which has a positive probability of being satisfied.

In this case, a sufficient condition for smoothness is that the discriminant $\det \begin{pmatrix} f_1 & f_2 & f_3 \\ f_2 & f_4 & f_5 \\ f_3 & f_5 & f_6 \end{pmatrix}$ is a squarefree polynomial. One can lower-bound the probability of this using the main Theorem 2.2 of On square-free values of large polynomials over the rational function field by Dan Carmon and viewing this discriminant as a polynomial in whichever of the $f_i$ has the largest degree, treating the other $f_i$ as fixed.

If the $d_i$ are close together, one could also use classical analytic number theory methods such as the circle method to get an estimate.