The easiest way to solve this is by identify the squares which are not connected to any other squares with names on them. (either vertically, horizontally or diagonally)
The first row (at the top) does not have any such squares.
The second row does not have any such squares.
The third row has two such squares. (the fifth and the last squares). Let us call them A and B.
The fourth row has one square. (the last square). Let us call it C.
The fifth has one. (the last square). Call it D.
The sixth has none.
The seventh has two. (the third and the fourth squares). Call it E and F.
The eighth has one. (the seventh square). Call it G
The ninth has none.
The tenth has none.
We must also now remember that the two squares that will be chosen must not touch each other.
1. A and B
2. A and C
3. A and D
4. A and E
5. A and F
6. A and G
7. B and D
8. B and E
9. B and F
10. B and G
11. C and E
12. C and F
13. C and G
14. D and E
15. D and F
16. D and G
17. E and G
18. F and G
(E and F), (B and C) and (C and D) cannot be pairs as these squares touch each other.
So there are 18 pairs available.