☆ Janine
Janine and Falkner seem to have quite a bit in common. What stands out the most between them is that they are both young gym leaders who have acquired their gym from their parents. And they both can't seem to find said parents. In the Generation IV remakes, you can find Falkner and Janine arguing over whose father is the best trainer. I don't know if their relationship extends beyond this, but it is quite possible considering that they intentionally meet up, which must mean they have each other's phone numbers in order to arrange such meetings. I don't know if their rivalry might get in the way of their relationship developing further or if it would help to cement their bonds.

☆ Morty
Falkner seems to be on speaking basis with all of the gym leaders, but he seems to be closer with Morty of the Ecruteak gym. In the manga, when Morty is being overwhelmed by people who wish to use him for his clairvoyant powers, Falkner is among the trainers who stand at the doors to the gym to defend him and allow him to rest. He battles with Silver to keep him out until it is known that Silver is allowed to enter, and it is he who accompanies him in and delivers him to Morty. There is a lot of speculation as to whether or not the two of them are something more, but it is not necessarily based on anything truly definitive.