No. Given any set of diophantine equations $f_1(z_1, ldots, z_n) = ldots = f_m(z_1, ldots, z_n)=0$, we can rewrite in terms of linear equations and quadratics. Create a new variable $w_{k_1 cdots k_n}$ for each monomial $z_1^{k_1} cdots z_n^{k_n}$ which occurs in the $f$'s, or which divides any monomial which occurs in the $f$'s. Turn each $f$ into a linear equation: For example, $x^3 y^2 + 7 x^2 y=5$ becomes $w_{32} + 7 w_{21} = 5$. Then create quadratic equations $z_i w_{k_1 cdots k_i cdots k_n} = w_{k_1 cdots (k_i +1) cdots k_n}$. For example, $x w_{22} = w_{32}.$ This shows that the solvability of Diophantine equations is equivalent to that of Diophantine equations of degree $leq 2$.
I'll also mention a very concrete case. The intersection of two quadrics in $mathbb{P}^3$ is a genus $1$ curve. To my knowledge, no algorithm is known to test for the existence of rational points even in this case. (But my knowledge is not very large.)
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