Alt--Basically Av allows you to manually set the aperture or lens opening, controlling the amount of light hitting the sensor and Tv let's you set the shutter speeds or how long that light hits the sensor for. When you set one funtion (high or low) the camera automatically adjusts the other setting (low or high). If you set higher f/stops less light is delivered so the camera has to set slower shutter speeds, causing blur with movement in low light and/or at low ISOs.
You can bump up the ISO settings for increased camera sensitivity to light but the results get "noisier" when you do, making things look "grainy". Above ISO 200 or even at ISO 200 you might not like the results in an enlargement. You could add more light but even an external strobe doesn't punch out all that far, maybe 5-6 feet at best. The simple answer is yes, go to Tv and dial in a higher shutter speed. (1/125th or 1/250th of a sec. would be great), assuming there is enough light to do this.
Now, you could stay in Av to also accomplish this by simply going to the lowest aperture number (allowing the most light in). The camera would then automatically compensate by setting the highest shutter speed it could for the situation. Yes, you do lose the added depth of field (more things in focus at varying distances) shooting at a wide open aperture but if it is a single fish, who cares about the background? If it's a school of fish at least many will be sharp at the distance you are focused. As Oly5050 mentioned, you can also follow the movement (panning) of a single fish to keep the main subject sharper.
Try to pay attention to what both the exposure settings are, at least until you start getting results that please you. This is all MUCH easier to learn above water, so go play with the camera and
practice. Just remember there is less light underwater as you go deeper, so practicing on a bright sunny day above water then going deep to shoot will increase the challenges.
I would also urge you to check the EXIF settings on your pictures (dig into your software to get the info displayed). There is a wealth of info recorded for each shot and knowing the actual f/stop shutter speed combos that were used for shots that worked or didn't work will help a lot in understanding how to improve things. You won't need all the info in the EXIF data but at least see what aperture and shutter speed settings were used.
Your mistakes can provide excellent learning tools--not because you keep making them but because you make the effort to
learn from them! Oh, did I mention PRACTICE?

// ww
Note to self: Must not sit at keyboard while having morning coffee...