Stability of geometrical isomers in cycloalkanes
To answer this question, you needs to know  what are the values for each acting forces on the cyclohexane frame work. Without that information, nobody would know if the repulsion forces (steric) would or would not be the dominating factor over hyrdogen bonding (electronic). Once we know the answer for that, you'd be able to calculate and find out the most stable structure. For your convenience, I attached a diagram with the chair conformations of all four structures and their Newman projections to make you understand the hydrogen bonding capabilities:

   

As you see, if the substitutions are not capable of hydrogen bonding, it is obvious that 1,3-cis structure is the most stable considering only stecic factors (eq-eq conformer). 1,3-trans structure (eq-eq conformer) is also stable relative to others but has an additional gauche interaction of vicinal bulky groups (1,2-substitutions) compared to its 1,3-counterpart.

In your case, however, all four molecules contain hydroxyl groups, so that three of them could be able to have structure stabilizing hydrogen bonding (the exception is trans-cyclohexane-1,3-diol; see Newman projection of IV-eq-ax). Both cis- and trans-cyclohexane-1,2-diols are structurally similar to 1,2-ethanediol, which favors a gauche conformation instead of an anti conformation, because only the gauche conformation allows hydrogen bonding between the two vicinal hydroxyl groups. Thus, both cis- and trans-cyclohexane-1,2-diols would adopt gauche conformations (see Newman projections of I-eq-ax and II-eq-eq) Ref.1. Keep in mind that, for cis-cyclohexane-1,2-diol (I), the anti conformation is impossible because of the restriction of ring structure. In trans-1,2-cyclohexanediol (II), however, the anti conformation is physically possible, but it would require the hydroxyl groups to be in diaxial positions, which is much less less stable than diequatorial due to 1,3-interactions imposed by axial 3,5-hydrogens (ax-ax chair confirmation).  According to the literature (Ref.1), cis-cyclohexane-1,2-diol is more acidic than trans-cyclohexane-1,2-diol by about $\pu{1.5 kcal/mol}$, hence 1,2-cis-compound would impose stronger hydrogen bonding than its trans-counterpart. 

According to the same study (Ref.1), cis-cyclohexane-1,3-diol is $\pu{10.9 kcal/mol}$ more acidic than trans-cyclohexane-1,3-diol, thus cis-compound is highly stable than its trans-counterpart (also remember, trans-cyclohexane-1,3-diol cannot have hydrogen bonding (vide spupra)). However, between conformers, the diaxial and diequatorial conformations in cis-cyclohexane-1,3-diol are relatively close in energy, so it is not immediately clear which conformation is dominant (Ref.1). Yet, a solution NMR study of this molecule showed that the diequatorial conformer (III-eq-eq) is more stable than diaxial conformar (III-eq-eq) Ref.2, although  III-eq-eq is impossible to make hydrogen bonding (see Newman projections of III-eq-eq and III-ax-ax). This shows that hydrogen bonding itself cannot overcome the steric repulsions induced by the 1,3-ax-ax-interactions. My conclusion is without knowing values of these attraction and repulsion interactions, it is impossible to say which of the three structures, I, II, and III), is the most stable.

References


X. Chen, D. A. Walthall, and J. I. Brauman, Acidities in Cyclohexanediols Enhanced by Intramolecular Hydrogen Bonds, J. Am. Chem. Soc., 2004, 126(39), pp. 12614–12620.
R. J. Abraham, E. J. Chambers, W. A. Thomas, Conformational analysis. Part 21. Conformational isomerism in cis-cyclohexane-1,3-diol, J. Chem. Soc., Perkin Trans. 2, 1993, (6), pp. 1061–1066.