(written from a Production point of view)
The Monterey Bay Aquarium is a public aquarium located south of the city of San Francisco in Monterey, California. The aquarium contains various aquatic life to include fish, octopus, penguins and sea otters. The aquarium does not host large marine mammals such as dolphins and whales.
Monterey Bay Aquarium was used as a filming location in Star Trek IV: The Voyage Home, specifically to show where the fictitious humpback whales, known as "George and Gracie", were located. For the film, the front of the aquarium (to include its logo) was modified with a sign, reading "Cetacean Institute," hung over the aquarium's name. Filming took place at the Aquarium on Thursday 24 April 1986, Monday 28 April 1986 and Wednesday 30 April 1986, among other dates.
To show the whale tank, the aquarium's tide water exhibit was filled in with sea water (this entire area of the aquarium is normally a dry bed with minimal water) and then the tide water tank was drained out for the dramatic scene where Gillian Taylor comes to work and finds that the whales have been taken without her knowledge. 
In Star Trek IV: The Voyage Home, James T. Kirk and Spock ride a bus across the Golden Gate Bridge to get to the Cetacean Institute, which the film states is located north in the city of Sausalito. In actuality, Monterey Bay Aquarium is almost a two hour drive south of San Francisco over state highways.