I found a question whether there are general methods to solve second degree Diophantine equations. I was unable to find an answer so is this known? In particular, the original writer wants to know whether one can find all integers satisfying $x^2 + x = y^2 + y + z^2 + z$.
About algorithm: There is an algorithm that will determine, given any quadratic $Q(x_1,\dots,x_n)$ as input, whether or not the Diophantine equation $Q(x_1,\dots,x_n)=0$ has a solution. This is something that I (and others) observed quite a long time ago. I have no knowledge about a nice algorithm.
Set one machine $M_1$ to search systematically for solutions. Another machine $M_2$ simultaneously checks whether there is a real solution (easy) and then checks systematically for every modulus $m$ whether there is a solution modulo $m$.
By the Hasse Principle (which in this case is a theorem), if our equation has "local" solutions (real and modulo $m$ for every $m$) then it has an integer solution. So either $M_1$ will bump into a solution or $M_2$ will find a local obstruction to a solution. Thus the algorithm terminates.
The corresponding question for cubics is unsolved. The same question for quartics (in arbitrarily many variables) is equivalent to the general problem of testing a Diophantine equation for solvability, so is recursively unsolvable.
Added: I think that the details are written out in the book Logical Number Theory I by Craig Smorynski. Very nice book, by the way.