= Solution
Use the usual <Feynman rule> convention in which the diagonal quadratic terms of the free <Lagrangian>, including the physical <W boson> and <Z boson> masses, are incorporated into their <propagators>. Then there are
$$
\boxed{0\text{ fundamental two-gauge-boson interaction vertices}.}
$$
A line joining two ends is a <propagator>, not an interaction vertex. <Gauge boson self-interactions> begin with cubic terms in the <Yang-Mills theory> <Lagrangian>. The question's $g,W,Z,\gamma$ name four families: they include eight <gluon> colour components and the two charged <W bosons> $W^\pm$.
There is a convention to distinguish here: if a calculation expands around massless free fields, quadratic $m_W^2W^+_\mu W^{-\mu}$ and $m_Z^2Z_\mu Z^\mu/2$ terms may be drawn as two-leg mass insertions. Those two insertion types belong to that bookkeeping convention. Similarly, two-point <self-energy> corrections occur in <loop diagrams>. Neither is an additional fundamental interaction in the standard physical-mass, tree-level rules used for the count above.
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