The Faq Gap is a break of Mountains or High Mountains in the ring of Impassable Mountains surrounding the Kingdom of Faq. Through this hex, a group of flying units and possibly mountain-capable units may enter or exit the kingdom. It forms at least one of the natural choke points into and out of the kingdom. (Another being the Tunnels through the mountains).
"Jillian informed Ansom that Faq could only be accessed by air, or through tunnels. Unless she was exaggerating, not even Mountain Units like the Western Giants can reach Faq through the Gap on foot, which suggests that at least part of the Faq Gap is composed of High Mountains rather than ordinary Mountains. Perhaps mountain capable units can't pass through high mountain hexes on foot (unless they are also tunnelers).
Impassable Mountains are a Terrain Type which cannot be passed even by flying Units. It's clear that there must be at least two places in the Mountains surrounding Faq where fliers can pass, because fliers from Transylvito used to pass through Faq airspace within visual range of Faq's cities, on the way to somewhere else. That is why Jack had to veil the cities.
Jillian might have been speaking generally, or might not have known about mountain-capable units. However, Jillian traveled widely as a mercenary, so she could easily have encountered mountain units.
Faq was a completely secret kingdom for a long time. If mountain capable troops could enter Faq at any time, it's more difficult to see how this level of secrecy could be maintained.