System of quadratic Diophantine equations 4 3 x 2 + 4 3 x + 1...

We consider only the general question. It was proved by Matijasevich that there is no algorithm which, on input any Diophantine equation $P(x_1,x_2,\dots ,x_m)=0$, where $P$ is a polynomial with integer coefficients, will determine whether the equation has an integer solution.
Using a little trick that goes back to Skolem, given any Diophantine equation $P(x_1,x_2,\dots ,x_m)=0$, we can algorithmically produce a system $Q_i(y_1,y_2,\dots ,y_n)=0$, $i=1,\dots ,s$ of quadratic Diophantine equations such that the system has a solution in integers if and only if $P(x_1,x_2,\dots ,x_m)=0$has a solution in integers.
It follows that there is no algorithm which, given any system of quadratic Diophantine equations, will determine whether the system has a solution in integers.