Fresh monkfish are caught during Swan Song. During the quest, Arnold Lydspor asks players to refill the fishing colony's food stocks. Players must catch and successfully cook five fresh monkfish. Catching a few extra in case some burn while cooking is not a bad idea, as even players with 99 cooking may burn them. However it is not recommended that players train Fishing during the quest, since these quest-specific fish provide much less experience for catching and cooking until the quest is complete. If eaten, it says "It's so fresh it's got no flavour, and it doesn't heal many life points either."
Like regular monkfish, players need a Fishing Level of 62 to catch fresh monkfish, but using fishing potions at level 59 is possible. An Admiral pie also works at Fishing Level 57. A spicy stew with three doses of brown spice added can be used to boost from 57, but it isn't recommended, since it has a chance of lowering Fishing level by five and the chance of boosting it is so low, you're better off using Admiral Pies.