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The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in [15]. In the present article, we employ the recently developed F-method, [10], [11] to the couple of non-compatible Lie algebras $\mathrm{Lie}{G}_2\stackrel{i}{\hookrightarrow }so\left(7\right)$, and generalized conformal $so\left(7\right)$-Verma modules of scalar type. As a result, we classify the $i\left(\mathrm{Lie}{G}_2\right)\cap 𝔭$-singular vectors for this class of $so\left(7\right)$-modules.