I don't have any advice on the fishing part of that, but the reason people say it won't be possible to HQ cooking without other classes is because crafting relies on a wide range of skills. Each crafting class provides one skill at level 15, another at 37, and another at 50 that can be used cross-class on other crafting classes to increase your HQ success in one way or another. The wider the selection of these cross-class skills you have access to, the better your crafting success will be, especially as you get up into higher-level recipes.

(It's actually still possible to HQ even without any of these cross-class skills, but only when you get lucky with RNG. Once you start dealing with expensive materials, you may not want to risk them on an RNG gamble. You'll want to be able to HQ more reliably than that.)