This was all taken from xivdb.com someone posted up some nice info on there regarding fishing these fish up.

Endoceras
Western La Noscea- Isles of Umbra Southshore
Prerequisite- **Weather must transition from Clear or Fair into Overcast, Wind or Fog**
Gathering- 397+
Weather- Overcast, Wind or Fog
Time- 8:00pm-4:00am
Bait- Spoon Worm > Fullmoon Sardine (Mooch)

Namitaro
South Shroud- Urth's Gift
Prerequisite- **Must fish a Giant Takitaro to obtain Fisher's Intuition for 1 minute**
Gathering- 397+
Weather- Thunder or Thunderstorms
Time- Any
Bait- Topwater Frog while Fisher's Intuition is active
Note: Use Glowworms for the Giant Takitaro then switch to Topwater Frog and be quick about it, you only get 2~ casts.

Helicoprion
Southern Thanalan- Sagolii Desert
Prerequisite- **Weather must transition from Fog into Heatwave**
Gathering- 397+
Weather- Heatwave
Time- 8:00am-8:00pm
Bait- Sand Leech > Storm Rider (Mooch)
NOTE: Best possible window to catch it is with a 8am-4pm heatwave following Fog. The Fog into Heatwave during daytime is uncommon and sometimes spread out days apart. Overcast into Heatwave might also work but have never seen Overcast->Heatwave during daytime.


Shonisaurus
Coerthas Central Highlands- Witchdrop
Prerequisite- **Weather must transition from Clear or Fair into Blizzard**
Gathering- 381+
Weather- Blizzard
Time- Any
Bait- Hoverworm > Cloud Cutter (Mooch) > Mahar (Mooch)
NOTE: Recommend 320+ Perception(possible requirement). Mahar seems to have a very low chance to come out in HQ.

Kuno The Killer
Mor Dhona- The North Shards
Prerequisite- **Must fish a Canavan to obtain Fisher's Intuition for 23 minutes**
Gathering- 397+
Weather- Gloom
Time- 9:00am-12:00am
Bait- Honey Worm > Silverfish/Goldfish (Mooch) > Assassin Betta (Mooch) while Fisher's Intuition is active
NOTE: Possible to catch in any weather as long as Fisher's Intuition buff is still active. Prerequisite is Gloom only.

Nepto Dragon
Eastern La Noscea- Rhotano Sea (Sterncastle)
Prerequisite- **Must fish 3 Little Thalaos to obtain Fisher's Intuition for 5 minutes**
Gathering- 403+
Weather- Rain or Showers
Time- Any
Bait- Northern Krill while Fisher's Intuition is active