In FEA the term convergence is mainly used in 2 aspects - mesh convergence and solution convergence in a non linear iterative process. Mesh convergence study is done to find out how small or fine the mesh size must be so that the results are nearly independent of the mesh size. Secondly, for non-linear problems which mainly use an iterative procedure, the term convergence/convergence criteria is used to tell the solver to stop iterating once this criteria is met, i.e. the solution does not change significant as iterations continue. In cases where this criteria is not met the solution doesn't converge and the solver throws an error. In your case, the contact pressure would be influence by both the mesh and contact settings (like stiffness/penetration). It would thus be a good idea to make sure both are apt. Check this video
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