We divide the circular boundary of a hyperbolic lattice into four equal intervals and study the probability of a percolation crossing between an opposite pair as a function of the bond occupation probability p. We consider the {7,3} (heptagonal), enhanced or extended binary tree (EBT), the EBT-dual, and the {5,5} (pentagonal) lattices. We find that the crossing probability increases gradually from 0 to 1 as p increases from the lower p_{l} to the upper p_{u} critical values. We find bounds and estimates for the values of p_{l} and p_{u} for these lattices and identify the self-duality point p corresponding to where the crossing probability equals 1/2. Comparison is made with recent numerical and theoretical results.