How many gauche butane interactions are present in cis-1,2-dimethylcyclohexane?
While cyclohexane does have gauche interactions, they are irrelevant because chair cyclohexane is taken as the reference point. The axial methyl group in cis isomer 1 has two gauche interactions with the ring as illustrated in red in 1a and 1b. A third gauche interaction is between the two methyl groups in 2. By comparison the trans isomer 3 only has gauche interactions between the methyl groups.

The difference in the number of gauche interactions - namely, two -  between these epimers is approximated by the difference in their heats of combustion[1] which is $\pu{1.54 kcal/mol}$. [1,2 cis: $\Delta H^\circ_\mathrm{comb} = \pu{-1248.31 kcal/mol}$; 1,2 trans: $\Delta H^\circ_\mathrm{comb} = \pu{-1246.77 kcal/mol}$.] A typical value for a gauche interaction is $\pu{0.9 kcal/mol/interaction}$.
References:

Walter H. Johnson, Edward Prosen, and Frederick D. Rossini, Heats of Combustion and Isomerization of the Eight $\ce{C8H16}$ Alkylcyclohexanes. Research Paper RP1S12, Volume 39, July 1947. PDF via NIST