As this answer says, most musical instruments as we know them won't work underwater. You might see something completely new arise, but we should assume that people will first reach for what is easiest. And what's that, for underwater music? Vocalization, as noted in the linked answer.
There's no particular reason to believe that it would sound anything like earthly vocal music, though. Consider these factors:
The scale. The western scale, an octave consisting of 12 half-steps (with variable placement, called temperament) arose from the Pythagorean system, a set of intervals in low ratios (like 3:2) that sound pleasing to the ear. These ratios are easily demonstrated, and were probably derived using, stringed instruments whose sounds travel through air to the human ear. They also occur as harmonic overtones on wind and brass instruments. There's a lot of math theory behind this too, but that's the core.1
Your people, however, are underwater. They've never heard stringed instruments (unless they came up onto land to do so). Their ideas of what is considered consonant or aesthetic need not have anything to do with these ratios. Sound travels differently through water, the land-based reference instruments are not available, and I'm not even addressing biological differences in ear construction. But you have the opportunity for your underwater music to sound very different, and that can start with a different scale. There's no reason they couldn't have come up with the same scale through math, but you can plausibly change this to make your water-dwellers different.
Melody and rhythm. You've heard whale-song. It, um, doesn't sound like pop music. :-) The sounds are longer and the intervals between "notes" tend to be small (almost "stepwise", were you to impose a scale on that). The latter might be constrained by cetacean vocal cords, but the former is probably in part because sound travels differently through water,2 with more echoing. I think this will push music toward being slower-moving; early underwater music might sound a lot like plainchant. (Plainchant also tends away from large-interval jumps; it happens, but most movement is stepwise, and this plus the lack of rhythmic variation contributes to that "somber" sound you associate with it.)
Harmony. I can't prove it, but I think that because of the echos and distortion, from any particular vantage point underwater a note will "linger" longer. This constrains how you can harmonize it if you want the results to be consonant (for local values of "consonant"; this depends on your musicians' biology, for starters).
On land, if you pluck or strike a string and do not interfere with it (damping it, fretting elsewhere along its length, etc), it will ring for some time after the initial note. Those notes are still "in the air" when you (or other instruments) play other notes. As a hammer-dulcimer player I take this into account when deciding what ornaments or harmonies I can get away with; ones that rely more heavily on consonant pitches (e.g. other notes in the current chord) work better than ones that use dissonant notes. On the other hand, there are certain ornaments (like the roll or trill) that work because they use immediately-adjacent notes, so this can go both ways. I suggest that your underwater music will rely mainly on consonances for harmonizing but will also explore the possibilities of well-placed dissonance. Harmony can also use rhythmic variation, so long as its overall shape fits into that long-decay property of underwater music.
1 I admit that I am failing to consider eastern musical scales here. My background is in western music; I don't know a lot about others. My impression is that similar considerations apply to pentatonic and Arabic scales.
2 Sound actually travels faster underwater, not slower; the reason it sounds distorted when you're swimming is the echoing, plus any part that travels through air. (Thanks TimB for correcting me.)