No co-prime neighbors
MATL, 14 bytes

!TM1*Zdl2$Xdl-


This outputs a non-empty column vector of nonzero numbers as truthy, or a vector containing at least a zero entry as falsy.


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Verify all test cases. The footer code contains an if branch to test truthyness/falsyness.
Since MATL uses ASCII, the source code is encoded as

[33 84 77 49 42 90 100 108 50 36 88 100 108 45]


which satisfies the requirement.


Explanation

!     % Implicit input. Transpose
TM    % Push input to latest function again
1*    % Multiply by 1 (does nothing, but matches factors)
Zd    % Compute gcd with broadcast: matrix of gcd of all pairs
l     % Push 1
2$    % The next function will use 2 inputs
Xd    % Extract diagonal 1 (i.e. that below the main diagonal) from the matrix
l-    % Subtract 1 from each entry. Implicitly display