You will be able to catch frogs in Quina's mini game after he joins your party. This is technically missable as you don't have to go to Qu's Marsh or ever have Quina in your party.
After leaving Lindblum via the dragons gate, you will see some Chocobo tracks. Just beyond those is a swampy area. Enter that and you will be on a path with Moggster and Moggy on it. You know them from the tutorials. Take the path directly behind them. Go straight up and you will find Quina trying to catch frogs. I caught him a frog, so I will assume that you should do the same. Cross the bridge and head up. Head inside the building and speak to Quale. Quina joins your party and you can now play the frog hunting mini game.
Go back to the area with frogs, you should now see a golden frog amongst the others. Catching the frog is going to take some patience, but for me, it was the first frog I caught after the one for Quina. Wait in a zone i.e. either end of the bridge for the frogs to hop on dry land and quickly nip out and attempt to grab a frog by holding
to sprint while tapping
to catch your frog. Eventually, the golden frog will be yours.
Also, make sure you leave one male and one female from in the pond so that they reproduce faster while you're away. If you do not do this, the next time you return, there will only be 2 tadpoles in the pond. You will then need to wait for them to mature before they can mate. Leaving behind a male, female and a golden frog will help you with the catch 99 frogs achievement faster. Female frogs can be spotted by the black stripe on either side, male frogs do not have stripes.