Ah, fire breathing dragons. Assuming your dragons are large beasts capable of flight, they are likely not exactly known for swimming, but if they can (though I doubt it), then larger fish may be possible targets.
More likely, fishing would be done in a strategy similar to perhaps the Bald Eagle - however, the size of the potential target depends on the dimensions of the dragon and the types and sizes of fish in your constructed world. To sustain such a large beast, significant amounts of food will be necessary, so there must be targets large enough to be worth hunting in order for fish to be the primary source of protein.
As for plant matter, such a large beast would likely need equivalently large sources of plant matter - in particular, fruit and other plants with enough size and nutritional value to be worth searching for. Bears as omnivores consume grasses and seeds as well as fungi and fruit, but the size of a dragon likely makes some of those options inadequate.
Overall, while your concept is possible (assuming the dragon is large and capable of flight), the environment will need to be shaped around the dragon itself. Given that it is omnivorous, I highly doubt that fishing is a good usage of its time given that the plant matter it would consume would most likely be on land where other forms of protein are likely to be jumping about.
I hope this provides a satisfactory response.
