Of course a dragonborn can play a flute. And an oboe.*
Dragonborn wonder how humans, elves, etc. can play plucked string instruments. How do you pluck the strings with your weird nubby fingers???
*I played the first in elementary, middle, and high school band, and the latter for my junior and senior years. I can absolutely say that any creature with sufficient lip muscles to speak a humanoid language--able to articulate plosives like p and b, and fricatives like f and v--would be able to play a flute or oboe. It might be slightly more difficult for them than for other species, e.g. needing to learn fine lip control, but (as my joke above alludes to) it's really not that much different from a human having to learn fine fingertip control to play a plucked instrument. If my semi-expert opinion doesn't get you over Reality Being Unrealistic, just go with the suggestion about the phorbeia--dragonborn flute-players use a leather face-strap to help make a smaller channel of air for blowing.
And now I'm imagining a neo-noire setting with an all-dragonborn jazz band. Potentially one that adds some pyrotechnics (or...uh...pneumotechnics, I guess, to account for varying breath type!)