Quadratic Diophantine equation - Find all integer solutions I have a Quadratic  Diophantine equation.

I am sorry I didn't get how to show correctly. So I used a
picture

How can I find all integer solutions? Should I do something with Z(14)?
The integer equation $11 x^2 - 14 y^2 = 1$ is impossible. However, I do not see that a proof is available for a beginner. There is no simple proof with congruences. There is one using, in effect, continued fractions.

The principal genus of this discriminant is two forms,
$$ x^2 - 154 y^2 $$
$$ 2 x^2 - 77 y^2,  $$
and your form $11x^2 - 14 y^2$ is $SL_2 \mathbb Z$ equivalent to $2x^2 - 77 y^2$ and simply does not integrally represent $1.$

The other genus is their negatives,
 $$ -x^2 + 154 y^2 $$
$$ -2 x^2 + 77 y^2,  $$

Here are a list of the class group, Gauss-Lagrange reduced forms, and the Lagrange cycle of your form

    1.             1          24         -10   cycle length            10
    2.            -1          24          10   cycle length            10
    3.             2          24          -5   cycle length             8
    4.            -2          24           5   cycle length             8

  form class number is   4

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 11 0 -14

  0  form             11           0         -14  delta      0
  1  form            -14           0          11  delta      1
  2  form             11          22          -3


          -1          -1
           0          -1

To Return  
          -1           1
           0          -1

0  form   11 22 -3   delta  -7     ambiguous  
1  form   -3 20 18   delta  1
2  form   18 16 -5   delta  -4
3  form   -5 24 2   delta  12
4  form   2 24 -5   delta  -4     ambiguous  
5  form   -5 16 18   delta  1
6  form   18 20 -3   delta  -7
7  form   -3 22 11   delta  2
8  form   11 22 -3


  form   11 x^2  + 22 x y  -3 y^2 

minimum was   2rep   x = 5   y = 39 disc 616 dSqrt 24  M_Ratio  4.760331
Automorph, written on right of Gram matrix:  
2419  5148
18876  40171
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
If you are asking for $(x, y)$ pairs that solve the equation: $$11x^2-14y^2=1$$ you can use the on solver at here to show that there are no solutions in integers.