His clothes, the music player and so on might of course move with him (supposing they would be able to withstand the rapid acceleration and decelleration involved, which I doubt), but the music player would have to play the music much faster all of a sudden for it to appear the same.
It's entirely possible of course for Quicksilver to have a specially modified cassette player and/or recording specifically for that purpose. That is of course if Quicksilver actually speeds up, there'd be less of a problem if he slows everything else down…
Edit: Some math
If we assume that the bullets travel at $340\,\mathrm{m/s}$ and the shooters are about $5\,\mathrm{m}$ from the X-Men, it would take the bullets about $\frac{340\,\mathrm{m/s}}{5\,\mathrm{m}}=0.1471\,\mathrm{s}$ to hit their targets. If we now assume that Quicksilver moves about $100\,\mathrm{m}$ in this time we arrive at a speed of $v=\frac{100\,\mathrm{m}}{0.0147\,\mathrm{s}}=6802.7211\,\mathrm{m/s}$, or about 6.8 km/s. Plugging that into the equation for the Lorentz factor $$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$ we arrive at $\gamma=1.00000000026$, which is pretty tiny.
However we have to consider who is actually moving here and who is at rest. Since Quicksilver is moving he is the one actually slowing down relative to the room (as in the twin paradox). That means that in the end (when he arrives back at the start) his clock will have slowed down relative to the room, not the other way around.