In the Hornblower series, Royal Navy series don't fish for provision. Is this an accurate depiction, and if so, why didn't the Navy fish for their food more often?

All Royal Navy ships in the days of sail carried fishing gear, including tackle and lines as well as trawl and seine nets. And it was policy to encourage fishing wherever it was practicable.

There’s an important distinction to be made between preserved fish and fresh fish. In the early days, salted fish was a standard part of ship’s provisions. Samuel Pepys, who worked for the Navy Board, wrote victualing contracts for navy ships dating from the 1670’s, which mention fish being served 3 days a week (Wednesdays, Fridays and Saturdays) in the form of “salt cod or hake”. But this practice was abandoned around the 1730’s, for several reasons:

Salted fish has a tendency to get damp and putrid. British sailors didn’t have a taste for rotten or “fermented” fish, unlike some other European navies.
There was a belief that salted foods, and especially salted fish, caused scurvy.
It was extremely smelly. In fact, navy ships used to have a separate cabin called the “fish room” so it could be stored away from everything else and the smell confined. Even after ships stopped carrying salted fish, the “fish room” remained, now used as storage for coal or other dirty items.
Many of the countries the Royal Navy fought were Catholic, and they traditionally served fish on Fridays and through Lent. So there was an association between fish and Catholicism, which didn’t endear fish to British sailors.

The situation was different for fresh fish, which the sailors enjoyed. The biggest thing in its favor was that you could get fresh fish while at sea, whereas all other meats came preserved from a barrel (or from cans, after canning became popular in the mid 19th century). Fresh fish made a nice change from preserved foods. In fact, it was Royal Navy policy (also mentioned in the Hornblower books) to not mess with enemy fishing boats. Instead, navy ships would stop and buy fresh fish from them, and possibly extract any useful military information if their captains could be bribed or fooled into revealing it.

As for doing their own fishing, it was common enough but difficult. Most of the open ocean is actually a desert, there aren’t many fish around. Fishing spots are usually in coastal waters, or in special areas where there are submerged undersea mountains that cause upwelling of the currents. But coastal waters were dangerous, enemy ships might be close by, and you don’t want to be caught towing a net when they chase you.

The war mission took precedence of course, but navy ships did stop and fish when they could. And on the open ocean you can still sometimes find bonito or porpoises following your boats, or sharks, or flying fish. All of these were caught and eaten as opportunity presented.

But fresh fish were insufficient to feed the crew on a regular basis. They were not counted as part of daily rations, they were treated as a surplus over and above the day’s ration. That’s one reason why fishing isn’t often mentioned in ships’ logs. It was something of an extra-curricular activity. There were guidelines on what to do with the limited catch. It went to the sickbay first, then any remainder was distributed among the crew.

Most of this information is from the book Feeding Nelson's Navy: The True Story of Food at Sea in the Georgian Era by Janet MacDonald.