$x^2-Dy^2=1$ has integer solution, does $x^2-Dy^2=-1$ also have? 
If I know that $x^2-Dy^2=1$ has integer solutions, what are the conditions that $x^2-Dy^2=-1$ has also integer solutions ?

For example $x^2-141y^2=1$ has a solution $(95,8)$ but for $x^2-141y^2=-1$ there doesn't exist a solution in integers.

The two questions are unrelated.  The first is called Pell's equation, and has solutions exactly when $D$ is not a square (apart from the trivial solution $x=1$, $y=0$).
The latter is called the negative Pell's equation.  It is the subject of ongoing study, some results of which may be found in that link. For example, if some prime $p|D$, and $p\equiv 3\pmod{4}$, then there is no solution for $D$. (Note: this explains $D=141=3\times 47$).  A list of those $D$ for which this is solvable can be found on OEIS. 
